==== TABLE OF CONTENTS ====
A-singular-normal-affine-surface-with-free-tangent-sheaf//01-introduction.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//02-curve.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//03-trace.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//04-jets.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//05-analytic.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//06-algebraization.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//bibliography.tex
A-singular-normal-affine-surface-with-free-tangent-sheaf//paper.tex
===========================

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//01-introduction.tex ====
\begin{abstract}
We construct a singular normal affine complex surface whose tangent
sheaf is free of rank two. This disproves the Lipman--Zariski conjecture
in characteristic zero.
\end{abstract}

\section{Introduction}

For a finitely generated integral $\CC$-algebra $A$, write
\[
\Der_{\CC}(A)=\Hom_A(\Omega^1_{A/\CC},A).
\]
The module $\Der_{\CC}(A)$ consists of the algebraic vector fields on
$\Spec A$. On a smooth variety of dimension $d$, the tangent sheaf is
locally free of rank $d$. The Lipman--Zariski conjecture asks for the
converse: a complex algebraic variety with locally free tangent sheaf
should be nonsingular. In affine terms, it asserts that local freeness
of $\Der_{\CC}(A)$ forces $A$ to be regular. We prove the following.

\begin{theorem}\label{thm:main}
There exist a finitely generated normal integral $\CC$-algebra $A$ of Krull dimension two and a maximal ideal $\m\subset A$ such that
\[
\Der_{\CC}(A)\simeq A^{\oplus2},
\qquad A_{\m}\text{ is not regular}.
\]
\end{theorem}

Here freeness holds on the entire affine surface, which can be chosen
smooth away from the marked point. We first construct an isolated
Gorenstein analytic surface singularity. The passage to algebraic geometry
preserves its completed local ring, from which tangent freeness descends.

\subsection{Context and significance}

Lipman's study of free derivation modules established that local freeness
of the tangent sheaf forces normality~\cite[Theorem~3]{Lipman1965}.
The conjecture thus asks whether the same hypothesis excludes the remaining
normal singularities. In positive characteristic the corresponding assertion
fails even for hypersurfaces: over a perfect field of characteristic $p$,
the singular surface $xy-z^p=0$ has tangent sheaf freely generated by
$x\partial_x-y\partial_y$ and $\partial_z$
\cite[\S1]{GrafPositive2022}. The construction here has a different source
of tangent freeness and takes place entirely over $\CC$.

Several positive results explain both the strength of tangent freeness
and the role of surfaces in the problem. Hochster proved that a reduced
finitely generated characteristic-zero algebra with a nonnegative grading
and degree-zero part equal to the base field is a polynomial algebra if
its derivation module is free at the irrelevant maximal ideal
\cite[\S1, Theorem]{Hochster1977}. Van Straten and Steenbrink used
extension of differential forms to prove the isolated-singularity case
in dimension at least three~\cite[Theorem~1.3 and~\S1.6]{VanStratenSteenbrink1985}.
Their extension argument leaves the surface case as a distinct problem.

Hypersurfaces were treated by Scheja and Storch~\cite[Satz~9.3]{SchejaStorch1972};
K\"allstr\"om established the characteristic-zero algebraic locally complete-intersection
case~\cite[Corollary~1.3(1), corrected version]{Kallstrom2011}.
Extension theorems for differential forms also give strong results defined
by the singularities of the canonical divisor. Greb, Kebekus, Kov\'acs and
Peternell proved the conjecture for Kawamata log terminal spaces
\cite[Theorem~6.1]{GKKP2011}. The log-canonical case was proved by
Druel~\cite[Theorem~1.1]{Druel2014} and by Graf and Kov\'acs
\cite[Corollary~1.3]{GrafKovacs2014}.
These results relate tangent freeness to the behavior of differential
forms on a resolution. Another distinction concerns the Lie bracket:
J\"order proved smoothness for a normal complex space whose tangent sheaf
locally admits a basis of pairwise commuting vector fields
\cite[Theorem~1.1]{Jorder2014}. Thus no tangent frame near our singular
point can be chosen commuting.

For normal complex surfaces, the cohomology and configuration of the
exceptional curves give further criteria. Graf proved smoothness when
the geometric genus $p_g$ is at most one, and also when $p_g=2$ and the
exceptional divisor is not a tree of rational curves
\cite[Theorem~1.1]{GrafLowGenus2021}. Bergner and Graf then obtained the
criterion
\[
 p_g-g-b\le2,
\]
where $g$ is the sum of the genera of the exceptional curves and $b$ is
the first Betti number of the resolution's dual graph
\cite[Theorem~1]{BergnerGraf2020}.
The analytic singularity constructed here has one exceptional curve of genus $21$
and self-intersection $-10$. Its canonical discrepancy is $-5$, and
$p_g\ge33$, so $p_g-g-b\ge12$. Remark~\ref{analytic:invariants}
proves these calculations. They place the analytic example beyond the
log-canonical and geometric-genus criteria just described.

Global hypotheses provide other positive cases. Biswas, Gurjar and Kolte
prove smoothness for a normal affine surface whose tangent bundle on
the smooth locus is trivial and whose smooth locus has logarithmic
Kodaira dimension at most one~\cite[Theorem~0.1(1)]{BiswasGurjarKolte2014}.
Consequently, the smooth locus $U$ of the affine surface in
Theorem~\ref{thm:main} has logarithmic Kodaira dimension two
(Corollary~\ref{algebra:log-kodaira}). Here logarithmic Kodaira dimension
means the Iitaka dimension of $K_{\overline U}+D$ on a smooth projective
compactification with reduced simple-normal-crossing boundary $D$.
This global consequence uses the algebraic endpoint of the construction,
not just the analytic germ. In contrast, Graf's compact-surface result
identifies compact complex surfaces with free rank-two tangent sheaf as
complex two-tori~\cite[Corollary~1.3]{GrafLowGenus2021}.

The example also gives a negative answer in characteristic zero to the
derivation-blowup question of Barajas, Ch\'avez-Mart\'inez and
Romano-Vel\'azquez~\cite[Introduction, Question~1]{BarajasChavezRomano2026}.
The blowup at a module is the universal proper birational modification
making its pullback modulo torsion locally free. As those authors observe,
a singular variety with free tangent sheaf cannot be resolved by iterating
this operation at the module of derivations: every step is the identity.
For our normal surface, inserting normalization does not change this
conclusion (Corollary~\ref{algebra:derivation-blowups}).

The neighborhood of the exceptional curve is essential to the argument.
Hochster's theorem excludes obtaining the example from a singular
positively graded cone with free tangent sheaf. Instead, we start with a
split line bundle and change the gluing maps of its zero-section
neighborhood while preserving the normal bundle. This freedom supplies
the required vector fields. Their dual volume form has a fifth-order
pole along the exceptional curve, so holomorphic extension across that
curve is not an input to the construction.

\subsection{Main ideas and proof structure}

The construction first produces an analytic surface singularity and then
passes to an affine algebraic surface. The analytic step requires two
vector fields on a punctured neighborhood that remain linearly
independent everywhere. We build those fields on a resolution, where the
exceptional set is a smooth negative curve. Their determinant vanishes
along that curve to a prescribed order, and nowhere else near it.

Let $C$ be a smooth curve and $L$ a positive line bundle with
$L^{\otimes4}\simeq K_C$. Its dual bundle $L^{-1}$ provides the initial
neighborhood of $C$. In compatible base and fiber coordinates $(z,t)$,
that neighborhood carries the meromorphic two-form
\[
\sigma=t^{-5}\,dt\wedge dz.
\]
We seek two vector fields of the form
\[
v_i=t\bigl(A_i(t,z)t\partial_t+B_i(t,z)\partial_z\bigr),
\qquad i=1,2,
\]
with $\sigma(v_1,v_2)=1+O(t)$. The leading vertical coefficients form a
pair $r=(r_1,r_2)$ of sections of $L$ with six simple common zeros,
whose reduced divisor is denoted by $Z$. These common zeros create the
main compatibility problem: each requires a pointwise correction to make
the leading determinant coefficient nonzero along $C$, but the corrections must also satisfy
global residue constraints to glue the vector fields. The curve and trace
construction produces data satisfying both requirements.

\paragraph{A curve and normalized residues.}
Section~\ref{curve:section} constructs $C$ on $\PP^1\times\PP^1$, with
fourth-order contact at $Z$ with an auxiliary elliptic curve $\Gamma$.
A two-torsion relation on $\Gamma$ controls the section dimensions on $C$ and supplies
six vectors in a two-dimensional quadratic space. The resulting curve
has genus $21$, and $\deg L=10$.

Section~\ref{trace:section} transports these vectors to $C$ by a trace
pairing for a degree-four map $C\to\PP^1$. Residue duality compares
principal parts on the two curves and fixes the scalar and sign of the
pairing. Proposition~\ref{prop:residue-data} then gives a meromorphic
pair $P$ of sections of $L^2$ whose local expansions enforce the determinant
condition and whose residue sums cancel the gluing obstructions.

\paragraph{Finite vector-field jets.}
In Section~\ref{jets:section}, we modify the neighborhood by finitely
many holomorphic flows and choose compatible coefficients of the fields.
At order two the obstruction separates into a symmetric residue sum and
one antisymmetric sum; the normalized residue data cancel both.
Thereafter, we extend the vector fields and their desired determinant
as separate but compatible finite Taylor expansions. Choosing the
determinant coefficients before solving for the field coefficients leaves
the freedom needed to remove the remaining obstructions. Exactness of a
differential and cohomology vanishing finish the finite induction.
Proposition~\ref{prop:jets} supplies finite jets
with determinant $t^5(1+O(t))$ in ordinary coordinates, to sufficient
order for the holomorphic lifting below.

\paragraph{Holomorphic lifting and contraction.}
Section~\ref{analytic:section} realizes the finite transition maps as
a Hausdorff surface neighborhood $Y$ of $C$. Set
$\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the sheaf of vector fields
tangent to $C$. Grauert's criterion contracts the curve to a point
$x$ of a normal surface $X$~\cite{Grauert1962}. Cohomology of the
successive ideal quotients and the theorem on formal
functions~\cite{Grauert1960} give the cohomological vanishing needed to
lift these jets to holomorphic fields after shrinking, without requiring
an infinite sequence of neighborhood
changes to converge. Their determinant gives a frame on
$Y\setminus C\simeq X\setminus\{x\}$, which normality extends to a
free tangent sheaf on $X$. Adjunction proves that $x$ is singular.

\paragraph{The affine algebraic endpoint.}
Section~\ref{algebra:section} combines approximation with the finite
determinacy of equations together with their relations. The latter
method appears in Hironaka's equivalence theorem
\cite[\S3, Theorem~3.3]{Hironaka1969} and in Artin's algebraization
of isolated singularities~\cite[Theorems~1.10 and~3.8--3.9]{Artin1969}.
We prove the required finite-determinacy statement with a relation matrix:
the approximated equations and matrix retain their exact zero product.
This produces an algebraic local ring with the same completion as
$\OO_{X,x}$. A finite Jacobian presentation identifies its completed
derivation module with the continuous derivations of that completion.
Faithful flatness descends freeness and the local ring properties.
Finally, shrinking to one affine neighborhood gives the globally free
module in Theorem~\ref{thm:main}, while preserving the singular point.

\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}p{0.19\textwidth}p{0.73\textwidth}@{}}
\toprule
Notation & Meaning and normalization \\
\midrule
$C$, $Z$ & Smooth genus-$21$ curve and its six marked points. \\
$L$, $r$ & $\deg L=10$, $L^4\simeq K_C$; the two sections in $r$ have
common zero divisor $Z$. \\
$\Gamma$, $P$ & Auxiliary elliptic curve; meromorphic pair of sections
of $L^2$ supplying the residue identities. \\
$Y$, $\cI$, $\cF$ & Surface neighborhood of $C$; $\cI=\OO_Y(-C)$ and
$\cF=\cT_Y(-\log C)$. The jets lie in $\cF/\cI^9\cF$. \\
$\pi:Y\to X$ & Contraction of $C$ to the singular point $x$;
$C^2=-10$. \\
$A$, $\mathfrak p$ & Affine algebra and marked singular point;
\mbox{$\widehat{A_{\mathfrak p}}\simeq\widehat{\OO_{X,x}}$}. \\
\bottomrule
\end{tabular}
\caption{The main objects in the construction, in their order of use.}
\label{tab:notation}
\end{table}

\subsection{Conventions}

Line bundles on curves are written additively when convenient: $aL+bH$ means $L^{\otimes a}\otimes H^{\otimes b}$, and $H^q(L)$ means $H^q(C,L)$. A divisor and its associated line bundle may share a symbol. For two rows $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$, put
\[
X\wedge Y=X_1Y_2-X_2Y_1.
\]
The symmetric bilinear form associated with a quadratic form $Q$ is normalized by $\langle x,x\rangle=Q(x)$. A residue on an annular overlap means the contour integral divided by $2\pi i$. Logarithmic residue and adjunction use the convention that the normal differential comes first. All germs and varieties are over $\CC$. We use standard facts about algebraic curve cohomology and adjunction as in~\cite{Hartshorne1977}, and about coherent analytic sheaves and normal spaces as in~\cite{GrauertRemmert1984}.
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//01-introduction.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//02-curve.tex ====
\section{A curve with prescribed residue data}
\label{curve:section}

We use additive notation for line bundles.  If $B$ is a line bundle and
$D$ is a divisor, $B+D$ means $B\otimes\OO(D)$; restrictions of bundles
will occasionally be left implicit.  We first construct a marked curve
and a fourth root of its canonical bundle.  An auxiliary elliptic curve
will control the directions and the normalization of the residue data.
The output is collected in Proposition~\ref{prop:curve-data}: besides
$L^4\simeq K_C$, we need precise section dimensions and evaluation maps
at six marked points. These conditions will make the low-order gluing
obstructions computable.

\subsection{The elliptic model and its quadratic form}

Let $\Gamma$ initially be a smooth elliptic curve.  Choose a degree-two
line bundle $F$ and a line bundle $D_0$ of exact order four, and set
$S=F+D_0$.  Both $F$ and $S$ are generated by their two-dimensional
spaces of sections.  The product of their associated maps embeds
$\Gamma$ in
\[
  \Sigma=\PP^1\times\PP^1.
\]
Indeed, after choosing an origin on $\Gamma$, the two fiber involutions
have the form $p\mapsto a-p$ and $p\mapsto a+d_0-p$, where $d_0\ne0$
represents $D_0$.  Two distinct points cannot be interchanged by both
involutions, and a point cannot be fixed by both.  The product map is
therefore injective and immersive, hence is a closed embedding.  Its
two projection degrees show that its image has bidegree $(2,2)$.
We now also write $F=(1,0)$ and $S=(0,1)$ for the ruling bundles on
$\Sigma$.

On $\Gamma$ put
\[
  \Delta=2D_0,\qquad \Phi=F+\Delta,\qquad M=3\Phi.
\]
Thus $\Delta$ is nontrivial of order two, and
\[
  \Gamma|_\Gamma=2F+2S=4F+\Delta,
  \qquad M=3F+\Delta,\qquad 2M=6F.
\]
Consequently
\begin{equation}\label{curve:bundle-identities}
\begin{split}
  4M&=(2\Gamma+4F)|_\Gamma,\\
  2M-(\Gamma+F)|_\Gamma&=\Phi,\\
  (\Gamma+2F)|_\Gamma-2M&=\Delta.
\end{split}
\end{equation}
Fix isomorphisms in these identities compatibly.  In particular,
viewing $\Phi$ as $2M-(\Gamma+F)|_\Gamma$, the first identity induces
the identification $2\Phi=2F|_\Gamma$ that we use below.

Since $K_\Sigma+\Gamma=0$, choose a nowhere vanishing logarithmic
two-form $\Omega$ with pole divisor $\Gamma$.  Its residue
$\omega=\Res_\Gamma\Omega$ is a nowhere vanishing differential on
$\Gamma$.  Our residue convention places the normal differential
first: locally $\Res_{y=0}((dy/y)\wedge dz)=dz$.
Choose a basis $u=(u_1,u_2)$ of $H^0(\Sigma,F)$, and write
\[
  \eta_\Gamma=u_1\,du_2-u_2\,du_1
       \in H^0(\Gamma,K_\Gamma+2F).
\]
The expression is independent of the local frame for $F$: in that
frame it is the indicated coefficient differential times the square
of the frame.

There is a unique element of $\Sym^2H^0(\Gamma,\Phi)$ whose image under
multiplication is
\begin{equation}\label{curve:quadratic}
  -\frac{\eta_\Gamma}{\omega}\in H^0(\Gamma,2\Phi)
       =H^0(\Gamma,2F).
\end{equation}
It defines a nondegenerate quadratic form $Q$ on
$H^0(\Gamma,\Phi)^*$.  To see both existence and nondegeneracy, let
$\iota_F$ and $\iota_\Phi$ be the two degree-two involutions.  One has
$\iota_\Phi=t_\Delta\iota_F$, where $t_\Delta$ is translation by the
nonzero point of order two corresponding to $\Delta$.  The four
ramification points of $F|_\Gamma$, namely the fixed points of
$\iota_F$, are paired without fixed points by $\iota_\Phi$.  They form
two distinct reduced fibers of the $\Phi$-map.  Their divisor is
therefore the pullback of the zero divisor of a binary quadratic with
two distinct roots.  Scaling this quadratic gives exactly
\eqref{curve:quadratic}.  Uniqueness follows because pullback from the
surjective $\Phi$-map is injective on sections of $\OO_{\PP^1}(2)$.

For $q\in\Gamma$, evaluation in a local frame of $\Phi$ gives a
nonzero element of $H^0(\Gamma,\Phi)^*$, defined up to a nonzero scalar.
The corresponding projective direction will be denoted by
$[\operatorname{ev}_q]$.  By \eqref{curve:quadratic}, this direction is
isotropic precisely when $q$ is a ramification point of $F|_\Gamma$.
For any quadratic form in this section, we use its polar form
\[
 x\mathbin{\cdot}y=\frac{Q(x+y)-Q(x)-Q(y)}2,
 \qquad x^2=Q(x).
\]

The six vectors below encode both a pointwise normalization and a total
residue.  If $w=\sum_jw_j$ and $w_j^2+w_j\mathbin{\cdot}w=1$ at
each marked point, then
\[
 \sum_{j=1}^6(w_j^2-1)=-w^2.
\]
The curve we construct has $\deg L=10$.  Thus the choice $w^2=90$
will give the total $-9\deg L$ required by the order-two obstruction
in Section~\ref{jets:section}, while retaining all six pointwise
normalizations.

\begin{lemma}\label{curve:six-vectors}
In a two-dimensional nondegenerate complex quadratic space, one can
choose three distinct nonisotropic directions, avoiding any prescribed
finite set of projective directions, and nonzero vectors
$w_1,\ldots,w_6$ that use each of these directions twice, with the two
vectors in each pair equal, such that
\begin{equation}\label{curve:vector-conditions}
  w=\sum_{j=1}^6w_j,\qquad
  w^2=90,\qquad w_j^2+w_j\mathbin{\cdot}w=1
  \quad(1\le j\le6).
\end{equation}
\end{lemma}

\begin{proof}
Choose coordinates in which $Q(x,y)=xy$, so that
$(x,y)\mathbin{\cdot}(x',y')=(xy'+yx')/2$.  Set
\[
 h=\sqrt{90}>0,\quad R=\sqrt{1+h^2/4}>0,\quad
 b=\frac{2h}{R},\quad
 d(z)=\left(Rz-\frac h2,\frac Rz-\frac h2\right).
\]
Direct expansion gives
\[
 d(z)^2+d(z)\mathbin{\cdot}(h,h)=R^2-h^2/4=1.
\]
For a sufficiently small general nonzero complex number $\lambda$,
let $z_1,z_2,z_3$ be the roots of
\[
 X^3-bX^2+b\lambda X-\lambda=0.
\]
Vieta's formulas give
\[
 \sum_{i=1}^3z_i=b,\qquad
 \sum_{i=1}^3z_i^{-1}=b,\qquad
 2\sum_{i=1}^3d(z_i)=(h,h).
\]
Thus taking each $d(z_i)$ twice gives \eqref{curve:vector-conditions}.

For completeness, these vectors have the required distinct directions.
As $\lambda\to0$, two roots approach zero and the third approaches
$b$.  The roots are distinct for general $\lambda$: the cubic
discriminant has nonzero leading term $-4b^3\lambda$ at zero.
Near zero the projective ratio of the two entries of $d(z)$ is
\[
 \frac{z(Rz-h/2)}{R-hz/2},
\]
whose derivative at zero is $-h/(2R)\ne0$.  The two small roots
therefore give distinct nonisotropic directions.  At $z=b$ the entries
are $3h/2$ and $(R^2-h^2)/(2h)$, both nonzero, so the third limiting
direction is different from their common limit.  These properties
persist for sufficiently small general $\lambda$.  Finally apply
simultaneously to all six vectors an isometry
$(x,y)\mapsto(\nu x,\nu^{-1}y)$.  On nonisotropic projective
directions this multiplies the ratio by $\nu^2$.  Only finitely many
values of $\nu\in\CC^*$ are excluded by the prescribed finite set,
and all equations in \eqref{curve:vector-conditions} are preserved.
\end{proof}

Apply Lemma~\ref{curve:six-vectors} to the quadratic form $Q$, excluding
the branch values of the $\Phi$-map.  Each of its three directions is
then $[\operatorname{ev}_q]$ at two distinct points of $\Gamma$.
Label these points $q_1,\ldots,q_6$ so that
$[w_j]=[\operatorname{ev}_{q_j}]$, and put
\begin{equation}\label{curve:marked-divisor}
 Z=q_1+\cdots+q_6\in|3\Phi|=|M|.
\end{equation}
Choose a section $\beta$ of $M$ with divisor $Z$.
Nonisotropy implies that $F|_\Gamma$ is unramified at all six points.
Their $F$-values are not all equal, since a degree-two fiber cannot
contain six distinct points.

\subsection{Fourth-order contact on the quadric}

We have chosen the six points and their quadratic-space data on the
elliptic curve. We now construct the curve $C$ through those points.
Fourth-order contact will transfer the short evaluation schemes from
$C$ to $\Gamma$, where their section dimensions are easy to compute.

Write $\gamma$ for the section of $\OO_\Sigma(\Gamma)$ defining
$\Gamma$.  We choose a section
\[
 \kappa\in H^0(\Sigma,2\Gamma+4F),\qquad
 \kappa|_\Gamma=\beta^4,
\]
using the fixed identification in \eqref{curve:bundle-identities}.
Such sections exist: restriction to $\Gamma$ has kernel
$\OO_\Sigma(\Gamma+4F)=\OO_\Sigma(6,2)$, whose first cohomology
vanishes.  The affine space of choices is
\[
 \kappa_0+\gamma H^0(\Sigma,\Gamma+4F).
\]
A general choice defines a smooth curve $C$.  Indeed, the associated
linear system is without base points off $\Gamma$, so Bertini's
Theorem gives smoothness there.  On $\Gamma\setminus Z$ every choice
is nonzero.  At each $q_j$, global generation of
$\OO_\Sigma(6,2)$ allows the normal derivative to vary nontrivially.
Avoiding six proper affine hyperplanes makes all those derivatives
nonzero.  Their complements and the Bertini open meet in the same
irreducible affine parameter space; independence of the six evaluation
functionals is unnecessary.

More explicitly, near $q_j$ choose a local equation $y=0$ for
$\Gamma$, a coordinate $z$ on $\Gamma$ vanishing at $q_j$, and frames
in which $\kappa$ has coefficient $k(y,z)$ and $\beta$ has coefficient
$b(z)$.  The frames may be chosen compatibly with
\eqref{curve:bundle-identities}, so that
\[
 k(0,z)=b(z)^4,\qquad b'(0)\ne0,\qquad k_y(0,0)\ne0.
\]
The implicit function theorem expresses $C$ as $y=\varphi(z)$, where
\begin{equation}\label{curve:contact-expansion}
 \varphi(z)=-\frac{b'(0)^4}{k_y(0,0)}z^4+O(z^5).
\end{equation}
Thus the contact order is exactly four and
$\operatorname{div}_C(\gamma|_C)=4Z$, with the same six marked points
now regarded as points of $C$.

The bidegree of $C$ is $(8,4)$.  The sequence for its defining divisor,
together with $H^1(\Sigma,\OO_\Sigma(-8,-4))=0$, gives
$H^0(C,\OO_C)=\CC$.  Hence the smooth curve $C$ is connected and
irreducible.  Put
\begin{equation}\label{curve:line-bundles}
 H=F|_C,\qquad L=H+Z,\qquad\theta=2L,
\end{equation}
and let $\xi$ be the canonical section of $\OO_C(Z)$.
The divisor equality above specifies the isomorphism
$4\OO_C(Z)\simeq\OO_C(\Gamma)$ by $\xi^4\mapsto\gamma|_C$.
Together with adjunction, this gives
\begin{equation}\label{curve:canonical-root}
 4L=4H+4Z\simeq(\Gamma+4F)|_C\simeq K_C.
\end{equation}
We fix the last isomorphism by the following normalization.  For a
local section $s$ of $\Gamma+4F$, it is
\begin{equation}\label{curve:adjunction-normalization}
 s|_C\longmapsto
 \Res_C\left(\frac{s\gamma\Omega}{\kappa}\right),
\end{equation}
with the normal differential first.  This formula fixes the scalar in
\eqref{curve:canonical-root} for the later residue computations.

\subsection{Section counts and evaluation maps}

We give the jet calculation underlying all the special-divisor
dimensions.  The section spaces below are algebraic section spaces;
equivalently, they are holomorphic section spaces on these projective
curves, by the comparison theorem of Serre~\cite[Theorem~1]{SerreGAGA1956}.

\begin{lemma}\label{curve:adjoint-jets}
For $0\le j\le3$, set
\[
 \mathcal A_j=\Gamma+(4-j)F=(6-j,2),\qquad
 N_j=\mathcal A_j|_\Gamma=(8-j)F+\Delta.
\]
Restriction identifies $H^0(\Sigma,\mathcal A_j)$ with
$H^0(C,K_C-jH)$ and maps it onto $H^0(\Gamma,N_j)$.
For $1\le\ell\le4$, evaluation on the length-$\ell$ subschemes at
the six marked points has the same rank on $C$ as on $\Gamma$.
The ranks needed here are
\begin{equation}\label{curve:residual-table}
\begin{array}{c|c|c|c|c|c}
 j&\ell& h^0(\Gamma,N_j)&N_j-\ell Z
   &h^0(\Gamma,N_j-\ell Z)&\text{evaluation rank}\\ \hline
 0&1&16&5F&10&6\\
 1&2&14&\Phi&2&12\\
 2&2&12&\Delta&0&12\\
 3&2&10&-F+\Delta&0&10
\end{array}
\end{equation}
\end{lemma}

\begin{proof}
The restriction kernels on $\Sigma$ are respectively
\[
 \mathcal A_j-C=(-2-j,-2),\qquad
 \mathcal A_j-\Gamma=(4-j,0).
\]
Product cohomology on $\PP^1\times\PP^1$ gives $H^0=H^1=0$ for
the first kernel and $H^1=0$ for the second.  Adjunction therefore
gives the two assertions about restriction.

By \eqref{curve:contact-expansion}, the ideals of the two length-$\ell$
subschemes in the local surface are
\[
 (y-\varphi(z),z^\ell)=(y,z^\ell)\qquad(\ell\le4).
\]
Thus these are the same embedded subscheme, and restrictions of the
surface line bundle and its sections to them agree.  The two
surjectivity statements just proved imply equality of evaluation ranks.

On the elliptic curve, $\deg F=2$, $Z=3F+\Delta$, and $2Z=6F$.
These identities give every residual bundle in
\eqref{curve:residual-table}.  A positive-degree line bundle on an
elliptic curve has as many sections as its degree; a negative-degree
bundle has no section.  The degree-zero bundle $\Delta$ is nontrivial,
so both its zeroth and first cohomology vanish.  Subtracting the
residual section dimensions from $h^0(\Gamma,N_j)$ gives the stated
ranks.  In particular, the twelve-dimensional surface system for
$j=3$ has a two-dimensional kernel on restriction to $\Gamma$, and
the resulting ten-dimensional system injects into its length-twelve
evaluation space.
\end{proof}

\begin{proposition}\label{prop:curve-data}
The construction above produces a smooth connected projective curve
$C$, a reduced divisor $Z=q_1+\cdots+q_6$, line bundles $H,L,\theta$,
and a basis $u=(u_1,u_2)$ of $H^0(C,H)$ with the following properties.
\begin{enumerate}
\item The numerical and canonical data are
\[
 \begin{gathered}
 g(C)=21,\qquad \deg H=4,\qquad\deg L=10,\\
 L=H+Z,\qquad\theta=2L,\qquad K_C=4L,
 \end{gathered}
\]
where the last isomorphism is normalized by
\eqref{curve:adjunction-normalization} and $\xi^4\mapsto\gamma|_C$.
The curve $C$ has exactly fourth-order contact with the auxiliary
elliptic curve $\Gamma$ at every $q_j$.
\item The pencil $u$ defines a finite degree-four morphism
$f:C\to\PP^1$ unramified at every $q_j$, and the values $f(q_j)$ are
not all equal.  The zero divisor of $u_1$ can be chosen to be a reduced
fiber disjoint from $Z$.
\item The section dimensions are
\begin{equation}\label{curve:section-dimensions}
\begin{gathered}
 h^0(C,jH)=j+1\quad(0\le j\le3),\\
 h^0(C,L)=h^0(C,H+2Z)=2,\qquad
 h^0(C,\theta)=3,\qquad h^0(C,\OO_C(Z))=1,\\
 h^0(C,\theta+H)=6.
\end{gathered}
\end{equation}
More precisely,
\begin{equation}\label{curve:section-multiplication}
\begin{split}
 H^0(C,L)&=\xi H^0(C,H),\\
 H^0(C,H+2Z)&=\xi^2H^0(C,H),\\
 H^0(C,\theta)&=\xi^2H^0(C,2H).
\end{split}
\end{equation}
In particular $r=\xi u$ is a basis of $H^0(C,L)$ with simple common
zero divisor exactly $Z$, and every section of $\theta$ vanishes
twice along $Z$.
\item For every $k\ge2$, the evaluation map
\begin{equation}\label{curve:evaluation-surjectivity}
 H^0(C,(k+1)L)\longrightarrow H^0(Z,((k+1)L)|_Z)
\end{equation}
is surjective.
\item The auxiliary data on $\Gamma\subset\Sigma$, including the
compatible bundle identities, the logarithmic volume $\Omega$, and
the quadratic form $Q$, are as in \eqref{curve:bundle-identities} and
\eqref{curve:quadratic}; the divisor $Z$ belongs to $|M|$.
The six nonzero vectors $w_j\in H^0(\Gamma,\Phi)^*$ have the
projective evaluation directions at $q_j$, use three distinct
directions, and satisfy \eqref{curve:vector-conditions}, with
$w^2=90=9\deg L$.
\end{enumerate}
\end{proposition}

\begin{proof}
The existence and contact assertions were established above.  From
$C=(8,4)$ and $K_C=(6,2)|_C$, intersection on the quadric gives
\[
 \deg H=4,\qquad \deg K_C=6\cdot4+2\cdot8=40,
 \qquad g(C)=21.
\]
Since $\deg Z=6$, one has $\deg L=10$.  The morphism given by $H$ is
the first projection, restricted to the irreducible curve $C$, hence
is finite of degree four.  Tangency at the marked points identifies
its differential there with that of $F|_\Gamma$, which is nonzero.
Its marked values are the corresponding $F|_\Gamma$-values, already
seen not to be all equal.  Choose a regular fiber away from these
finitely many values.  A determinant-one change of the basis $u$
makes this fiber the zero divisor of $u_1$.  Such a change preserves
$u_1\,du_2-u_2\,du_1$ on both curves, and hence preserves the
quadratic form and every normalization already chosen.

For $0\le j\le3$, the sequence restricting $\OO_\Sigma(j,0)$ to
$C$ has kernel $\OO_\Sigma(j-8,-4)$, with $H^0=H^1=0$ by product
cohomology.  This proves $h^0(C,jH)=j+1$.  For any line bundle $B$
on $C$ and effective divisor $D$, Riemann--Roch and Serre duality give
\begin{equation}\label{curve:dimension-formula}
 h^0(C,B+D)-h^0(C,B)
 =\deg D-\operatorname{rank}\left(
 H^0(C,K_C-B)\longrightarrow H^0(D,(K_C-B)|_D)\right).
\end{equation}
Apply this with $B=jH$ and use the ranks in
Lemma~\ref{curve:adjoint-jets}.  Its first three rows give
\[
 h^0(C,\OO_C(Z))=1,
 \quad h^0(C,H+2Z)=2,
 \quad h^0(C,2H+2Z)=3.
\]
The inclusions of $H^0(C,H)$ into the spaces for $H+Z$ and $H+2Z$
then give $h^0(C,L)=2$.  The fourth row gives
\[
 h^0(C,3H+2Z)=h^0(C,3H)+12-10=6.
\]
This proves \eqref{curve:section-dimensions}, and equality of the
dimensions in the natural multiplication inclusions proves
\eqref{curve:section-multiplication}.  Because the pencil $u$ has no
base point and $Z$ is reduced, the assertion about the common zeros
of $r$ follows as well.

Finally, the divisor sequence for $N=(k+1)L$, followed by Serre
duality, shows that \eqref{curve:evaluation-surjectivity} is
surjective if and only if the natural inclusion
\[
 H^0(C,(3-k)L)\longrightarrow H^0(C,(3-k)L+Z)
\]
is an isomorphism.  For $k=2$ this is the inclusion from $L$ to
$L+Z=H+2Z$, between two-dimensional spaces.  For $k=3$ it is the
inclusion from $\OO_C$ to $\OO_C(Z)$, between one-dimensional
spaces.  For $k\ge4$ both spaces vanish, since the target bundle has
degree $10(3-k)+6\le-4$.  The final assertion follows from
Lemma~\ref{curve:six-vectors} and \eqref{curve:marked-divisor}.
\end{proof}

For later use, Riemann--Roch and Serre duality also give
\begin{equation}\label{curve:boundary-cohomology}
 h^1(C,L)=h^0(C,3L)=12,\qquad h^0(C,4L)=21.
\end{equation}
Indeed, $h^0(C,L)-h^1(C,L)=10+1-21=-10$ and
$K_C-L=3L$.

One additional consequence will be useful when removing a
two-dimensional obstruction.  In local frames compatible with
$L=H+Z$, with $\xi$ represented by a parameter $z$ at a marked point,
one has $r=zu$ and
$r'(0)=u(0)$.  The resulting six nonzero row vectors $r'(q_j)$ span
$\CC^2$: their projective directions are the pencil values $f(q_j)$,
which are not all equal.  Changing the frames or the local parameters
only rescales each of these rows.
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//02-curve.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//03-trace.tex ====
\section{The trace form and the residue data}
\label{trace:section}

The goal of this section is Proposition~\ref{prop:residue-data}: a
meromorphic pair with prescribed poles and two normalized residue
identities. The trace form translates the quadratic-space data on the
elliptic curve into those identities on $C$. The comparison must preserve
its scalar and sign, since both enter the first gluing obstruction.

Retain the data and the normalizations of Proposition~\ref{prop:curve-data}.
In particular, $\Omega$ is the fixed logarithmic volume on $\Sigma$, its
normal-first residue is the nowhere vanishing differential $\omega$ on
$\Gamma$, and $4L\simeq K_C$ is the specified adjunction isomorphism.
Put
\[
 \theta=2L,\qquad \Lambda=\theta+H=3H+2Z,
 \qquad f:C\longrightarrow\PP^1.
\]
For the chosen basis $u=(u_1,u_2)$ of the ruling pencil, write
$\eta_C=u_1\,du_2-u_2\,du_1\in H^0(C,K_C+2H)$.
Although the individual derivatives depend on a frame of $H$, this
alternating expression does not.  We use the analogous notation
$\eta_\Gamma$ on $\Gamma$.

\subsection{A two-dimensional quotient of a perfect trace form}

The pairing below is a concrete instance of finite-morphism duality,
as developed by Grothendieck and presented in Hartshorne's
\emph{Residues and Duality}~\cite[Chapter~III, \S6]{Hartshorne1966}. We check its local
perfection directly so that the trace normalization remains explicit.

\begin{lemma}\label{trace:pairing}
The rank-four vector bundle $E=f_*\Lambda$ has a perfect symmetric pairing
\begin{equation}\label{trace:form}
 E\otimes E\longrightarrow\OO_{\PP^1},\qquad
 (T_1,T_2)\longmapsto
 \Tr_{C/\PP^1}\!\left(\frac{T_1T_2}{\eta_C}\right).
\end{equation}
Its splitting type is $(3,0,0,-3)$.  Multiplication by $\xi^2$ identifies
$\OO_{\PP^1}(3)$ with its positive line subbundle $\mathcal P$.
The form on global sections has radical
$\xi^2H^0(C,3H)$, and induces a nondegenerate symmetric form
$\langle\ ,\ \rangle$ on
\begin{equation}\label{trace:W}
 W=H^0(C,\Lambda)/\xi^2H^0(C,3H),\qquad \dim W=2.
\end{equation}
For every $x\in\PP^1$, evaluation identifies this quadratic space with
$(\mathcal P^\perp/\mathcal P)_x$.

Suppose $q\in Z$, and choose local frames $\mathfrak f$ of $H$ and
$\tau$ of $\Lambda$, and a parameter $z$ at $q$.  Write, using the fixed
isomorphism $2\Lambda\simeq K_C+2H$,
\[
 \tau^2=s(z)\,dz\,\mathfrak f^2,
 \qquad \eta_C=\mathcal E(z)\,dz\,\mathfrak f^2.
\]
Coefficient evaluation $\ell_q:T\mapsto(T/\tau)(q)$ descends to $W$.
Its square in the dual form is
\begin{equation}\label{trace:dual-evaluation}
 \ell_q^2=\frac{\mathcal E(q)}{s(q)}.
\end{equation}
In particular, it is nonzero and nonisotropic.
\end{lemma}

\begin{proof}
The morphism $f$ is finite and flat of degree four.  The identity
$2\Lambda=K_C+2H$ identifies $\Lambda^2$ with the relative dualizing
bundle.  We verify directly the normalization and perfection of its trace
pairing.  Locally at a ramification point one can choose parameters with
$x=w^e$ on a single component over a target disk.  A frame of $H$ pulled
back from that disk can be chosen so that
$\eta_C=dx\,\mathfrak f^2$.  If
$\tau^2=a(w)\,dw\,\mathfrak f^2$, with $a$ a unit, the local pairing is
\[
 (b,c)\longmapsto
 \Tr\!\left(\frac{a(w)bc}{e w^{e-1}}\right).
\]
On the basis $1,w,\ldots,w^{e-1}$, the pairing without the unit $a$ has
matrix
\[
 \Tr\!\left(\frac{w^{i+j}}{e w^{e-1}}\right)
   =\begin{cases}1&i+j=e-1,\\0&i+j\ne e-1,
     \end{cases}
 \qquad 0\le i,j<e.
\]
It is therefore holomorphic and perfect.  Multiplication by $a$ is an
automorphism, and taking the direct sum over the points above the target
disk proves the assertion.  On an unramified fiber the trace is the sum
of its four contributions; there is no division by the degree.

By the splitting theorem for vector bundles on $\PP^1$
\cite[\S2, Theorem~2.1 and Corollary]{Grothendieck1957}, $E$ is a direct sum of line bundles.
Self-duality implies that its splitting degrees occur in opposite
pairs.  Proposition~\ref{prop:curve-data} gives
\[
 h^0(E(-1))=h^0(\theta)=3,\qquad
 h^0(E(-2))=h^0(H+2Z)=2.
\]
A positive splitting degree $a$ contributes $a$ and $a-1$, respectively,
to these two numbers, and all nonpositive degrees contribute zero.
There is consequently exactly one positive degree, equal to three.
Thus
\[
 E\simeq\OO(3)\oplus\OO^{\oplus2}\oplus\OO(-3).
\]
The nonzero map $\OO(3)\to E$ obtained from multiplication by $\xi^2$
has zero components in the three lower-degree summands.  Its component
in $\OO(3)$ is a nonzero constant.  Hence its image $\mathcal P$ is
the entire positive summand, and is a subbundle, including over the
images of the points of $Z$.
The injective pullback
$H^0(\PP^1,\OO(3))\to H^0(C,3H)$ is an isomorphism, since both
spaces have dimension four.  Thus
$H^0(\mathcal P)=\xi^2H^0(C,3H)$.

The line $\mathcal P$ is isotropic, since
$\Hom(\OO(6),\OO)=0$.  Likewise it pairs trivially with the two
degree-zero summands.  Perfection gives
\[
 \mathcal P^\perp\simeq\OO(3)\oplus\OO^{\oplus2},\qquad
 \mathcal P^\perp/\mathcal P\simeq\OO^{\oplus2},
\]
with a nondegenerate constant form on the quotient.  All global sections
of $E$ lie in $\mathcal P^\perp$, and $H^1(\mathcal P)=0$.
Taking sections proves the assertions about $W$ and its fibers.

At $q\in Z$, evaluation annihilates $\mathcal P_{f(q)}$, because the
map defining that line is multiplication by $\xi^2$.  The finite fiber algebra is a product of its Artin-local factors,
and the trace pairing is the orthogonal sum of their pairings.
Since $f$ is unramified at $q$, the factor at $q$ is one-dimensional,
with form coefficient $s(q)/\mathcal E(q)$. Evaluation at $q$
vanishes on the other factors, whether or not they are ramified.
The metric dual of coefficient evaluation thus has square
$\mathcal E(q)/s(q)$ in $E_{f(q)}$.  It lies in
$\mathcal P_{f(q)}^\perp$, because the evaluation kills
$\mathcal P_{f(q)}$.  Passing to the orthogonal quotient preserves this
square and gives the functional on $W$.  Finally
$\mathcal E(q)\ne0$ by the unramifiedness in
Proposition~\ref{prop:curve-data}.
\end{proof}

We write a dot for the dual bilinear form on $W^*$, so that
$\lambda^2=\lambda\cdot\lambda$.  The distinction between this dual
form and the form on $W$ accounts for the reciprocal in
\eqref{trace:dual-evaluation}.

\subsection{Identification by full principal parts}

The preceding lemma gives a quadratic space on $C$. We now identify it
with the one on $\Gamma$ used to choose the marked points. The
principal-part exact sequence, together with Serre duality
\cite[\S3, Theorem~4; \S4, Theorem~5]{SerreDuality1955}, characterizes
realizable principal parts by their residue pairings with global adjoint
sections.
We first compare both coefficients of each double pole to identify the
two spaces.  We then use the leading coefficients to determine the
scalar and sign of the quadratic form under that identification.

\begin{lemma}\label{trace:principal-parts}
There is an isomorphism
\begin{equation}\label{trace:iota}
 \iota:W\xrightarrow{\ \sim\ }H^0(\Gamma,\Phi)
\end{equation}
that identifies the full principal parts of $T/\xi^2$ on $C$ with
those of $\iota([T])/\beta^2$ on $\Gamma$ through residue duality.
It identifies the projective evaluations at corresponding points of $Z$.
Moreover, its dual
\[
 \iota^*:H^0(\Gamma,\Phi)^*\longrightarrow W^*
\]
is an isometry for the quadratic form $Q$ of
Proposition~\ref{prop:curve-data} and the dual trace form.
\end{lemma}

\begin{proof}
Set $N_\Sigma=\Gamma+F$, $N_\Gamma=N_\Sigma|_\Gamma$, and $D=2Z$.
Fourth-order contact identifies $D$ on $C$ and on $\Gamma$ as the same
embedded length-twelve subscheme of $\Sigma$.  The isomorphism
$N_\Sigma|_C\simeq K_C-3H$ is the one supplied by adjunction with the
fixed volume.  The differential $\omega$ trivializes $K_\Gamma$.
For a line bundle $B$ on a smooth curve, the local pairing between
principal parts of $B(D)$ and jets of $K-B$ is
\[
 (a,s)\longmapsto\sum_{q\in Z}\Res_q(as).
\]
It is perfect: in a parameter and mutually dual frames the two
coefficients of $a_{-2}z^{-2}+a_{-1}z^{-1}$ pair with a jet
$s_0+s_1z$ as $a_{-2}s_1+a_{-1}s_0$.
Consequently the two principal part spaces
\[
 H^0\!\left(C,\frac{\OO_C(3H+2Z)}{\OO_C(3H)}\right),
 \qquad
 H^0\!\left(\Gamma,
       \frac{\OO_\Gamma(-N_\Gamma+2Z)}{\OO_\Gamma(-N_\Gamma)}\right)
\]
are both identified by residues with
$H^0(D,N_\Sigma|_D)^*$.  This defines their comparison without any
choice of coordinates or frames. It compares the full local
principal-part spaces before any condition of global realizability
is imposed.

Both complete adjoint systems are restrictions of the same ambient
system $H^0(\Sigma,N_\Sigma)$.  Indeed, the kernels in the restriction
sequences to $C$ and to $\Gamma$ are respectively
\[
 \OO_\Sigma(-\Gamma-3F)=\OO_\Sigma(-5,-2),
 \qquad \OO_\Sigma(F)=\OO_\Sigma(1,0),
\]
and both have vanishing first cohomology.  Their images on the common
scheme $D$ therefore coincide.  Equivalently, an ambient section
vanishing on $\Gamma$ vanishes to order at least four on $C$ at $Z$
and has no effect on these residue tests.

The principal part exact sequence and Serre duality now say that the
image of $H^0(C,\Lambda)$, by $T\mapsto\operatorname{pp}(T/\xi^2)$,
is precisely the annihilator of this common adjoint jet space; its
kernel is $\xi^2H^0(C,3H)$.  On $\Gamma$ the same annihilator is the
image of
\[
 H^0(\Gamma,\Phi)\longrightarrow
 H^0\!\left(\Gamma,
       \frac{\OO_\Gamma(-N_\Gamma+2Z)}{\OO_\Gamma(-N_\Gamma)}\right),
 \qquad \widetilde T\longmapsto
       \operatorname{pp}(\widetilde T/\beta^2),
\]
because $\Phi=2M-N_\Gamma$ and $(\beta)=Z$.
This map is injective: its kernel would give a section of
$-N_\Gamma$, which has degree $-10$.
These facts establish \eqref{trace:iota}.  In particular, this is an
identification of full principal parts.  Its dimension can also be
read directly on $\Gamma$: $h^0(N_\Gamma)=10$, and restriction to
$D$ is injective since $\deg(N_\Gamma-2Z)=-2$; its annihilator has
dimension $12-10=2$.

We compute the local comparison, including its first derivative and the
adjunction scalar.  Near a point of $Z$ choose coordinates $(y,z)$ and
frames $\mathfrak g$ of $\Gamma$ and $\mathfrak f$ of $F$ such that
\[
 \gamma=y\mathfrak g,\qquad
 \Omega=e(y,z)\frac{dy}{y}\wedge dz,\qquad e(0,z)=1,
\]
so $\omega=dz$ on $\Gamma$.  Write
\[
 \kappa=k(y,z)\mathfrak g^2\mathfrak f^4,
 \qquad \beta=b(z)\mathfrak m,
 \qquad \mathfrak m^4=\mathfrak g^2\mathfrak f^4\big|_\Gamma.
\]
Here $k(0,z)=b(z)^4$ and
$b'(0)k_y(0,0)\ne0$.  With $K(z)=k_y(0,z)$, the equation of $C$ gives
\begin{equation}\label{trace:contact-expansion}
 y|_C=-\frac{b(z)^4}{K(z)}+O(z^8),\qquad
 e|_C=1+O(z^4),\qquad k_y|_C=K(z)+O(z^4).
\end{equation}
The normal-first adjunction convention sends the frame
$\mathfrak g\mathfrak f^4$ to
\begin{equation}\label{trace:adjunction}
 \Res_C\!\left(\frac{e(y,z)\,dy\wedge dz}{k(y,z)}\right)
       =\frac{e|_C}{k_y|_C}\,dz.
\end{equation}
The sign is positive here, since
$dy\wedge dz=(dk\wedge dz)/k_y$.

Let $a=\xi/z$ be the local frame of $\OO_C(Z)$ and
$\tau=\mathfrak f^3a^2$ the resulting frame of $\Lambda$.
The fixed fourth-power identification sends
$a^4$ to $(y|_C/z^4)\mathfrak g$.
Using \eqref{trace:adjunction}, its square has the form
\begin{equation}\label{trace:frame-square}
 \tau^2=s(z)\,dz\,\mathfrak f^2,
 \qquad
 s(z)=\frac{e|_C\,y|_C}{z^4 k_y|_C},\qquad
 s(0)=-\frac{b'(0)^4}{k_y(0,0)^2}.
\end{equation}
Thus the negative sign comes from solving the equation of $C$.

Write $T=\pi(z)\tau$, and write
$\widetilde T=\iota([T])$ in the frame
$\widetilde\tau=\mathfrak m^2/(\mathfrak g\mathfrak f)$ of $\Phi$
as $\widetilde T=\widetilde\pi(z)\widetilde\tau$.
For an ambient test section $A(y,z)\mathfrak g\mathfrak f$, the two
residue tests are
\begin{equation}\label{trace:local-residue-tests}
 \Res_0\!\left(
  \frac{A|_C\,\pi\,e|_C}{k_y|_C\,z^2}\,dz\right),
 \qquad
 \Res_0\!\left(
  \frac{A(0,z)\,\widetilde\pi}{b(z)^2}\,dz\right).
\end{equation}
The restrictions of $A$ to the two curves agree modulo $z^4$.
It follows from \eqref{trace:contact-expansion} that equality of their
residue functionals on all local test jets is exactly
\begin{equation}\label{trace:full-pp}
 \operatorname{pp}\!\left(\frac{\pi}{K(z)z^2}\right)
  =\operatorname{pp}\!\left(\frac{\widetilde\pi}{b(z)^2}\right),
 \qquad
 \widetilde\pi\equiv\frac{(b(z)/z)^2}{K(z)}\pi\pmod{z^2}.
\end{equation}
To display both coefficients, put
\[
 b=b_1z+b_2z^2+O(z^3),\quad K=K_0+K_1z+O(z^2),\quad
 \pi=\pi_0+\pi_1z+O(z^2).
\]
The respective principal parts are
\[
 \frac{\pi_0}{K_0}z^{-2}
 +\left(\frac{\pi_1}{K_0}-\frac{K_1\pi_0}{K_0^2}\right)z^{-1},
 \qquad
 \frac{\widetilde\pi_0}{b_1^2}z^{-2}
 +\left(\frac{\widetilde\pi_1}{b_1^2}
        -\frac{2b_2\widetilde\pi_0}{b_1^3}\right)z^{-1}.
\]
Hence
\begin{equation}\label{trace:two-coefficients}
 \begin{split}
 \widetilde\pi_0&=\frac{b_1^2}{K_0}\pi_0,\\
 \widetilde\pi_1&=\frac{b_1^2}{K_0}\pi_1+
       \left(\frac{2b_1b_2}{K_0}
                    -\frac{b_1^2K_1}{K_0^2}\right)\pi_0.
 \end{split}
\end{equation}
This invertible triangular transformation is the local expression of
the intrinsic residue-dual identification above; therefore its expressions
in different coordinates and frames agree.  In particular, no choice
of independent pointwise scalars is being substituted for a global map.

It remains to identify its metric.  In these frames write
$\eta_C=\mathcal E_C\,dz\,\mathfrak f^2$ and
$\eta_\Gamma=\mathcal E_\Gamma\,dz\,\mathfrak f^2$.
Tangency gives $\mathcal E_C(0)=\mathcal E_\Gamma(0)$.
Let $\ell$ and $\widetilde\ell$ be coefficient evaluations in $\tau$
and $\widetilde\tau$, respectively.  By
\eqref{trace:two-coefficients},
\[
 \iota^*\widetilde\ell=\alpha\ell,
 \qquad \alpha=\frac{b_1^2}{K_0}.
\]
Equations~\eqref{trace:dual-evaluation} and
\eqref{trace:frame-square} therefore give
\begin{equation}\label{trace:isometry-scalar}
 (\iota^*\widetilde\ell)^2
     =\alpha^2\frac{\mathcal E_C(0)}{s(0)}
     =-\mathcal E_\Gamma(0).
\end{equation}
The compatible bundle identifications give
$\widetilde\tau^2=\mathfrak f^2$.  Thus the final quantity in
\eqref{trace:isometry-scalar} is exactly $Q(\widetilde\ell)$, since
the section defining $Q$ multiplies to $-\eta_\Gamma/\omega$.
The evaluations at $Z$ have three distinct projective directions.
The difference of the two binary quadratic forms vanishes on those
three directions, and hence vanishes identically.  This proves the
isometry, with its sign and scalar fixed.
\end{proof}

\subsection{Meromorphic sections with the required residues}

The isometry now turns the six chosen vectors into prescribed pole
coefficients on $C$.  A global residue calculation will show that these
poles force the required local determinant normalization; no further
choice of regular parts will be needed.
We pass from the ambient coordinates used for adjunction to coordinates
adapted to $4L\simeq K_C$.  On a small disk centered at $q_j$ choose
a parameter $z_j$ and a frame $\ell_j^{\mathrm{fr}}$ of $L$ whose fourth
power maps to $dz_j$.  Such a frame exists by taking a holomorphic fourth
root of a unit.  Set
\[
 a_j=\xi/z_j,\qquad
 \mathfrak f_j=\ell_j^{\mathrm{fr}}/a_j,
 \qquad \tau_j=(\ell_j^{\mathrm{fr}})^2\mathfrak f_j.
\]
These are frames of $\OO_C(Z)$, $H$, and $\Lambda$, respectively.
If $r=\xi u$, its coefficient row in $\ell_j^{\mathrm{fr}}$ satisfies
$r=z_j u$, where here $u$ denotes its coefficient row in
$\mathfrak f_j$.  The fiber-linear function $t_j$ on $L^{-1}$
corresponding to $\ell_j^{\mathrm{fr}}$ gives compatible split
coordinates $(t_j,z_j)$.  Let $l_j\in W^*$ be coefficient evaluation
in $\tau_j$.  Since $\tau_j^2=dz_j\,\mathfrak f_j^2$,
Lemma~\ref{trace:pairing} reads in these coordinates as
\begin{equation}\label{trace:compatible-norm}
 l_j^2=(u_1u_2'-u_2u_1')(0).
\end{equation}
Primes in the rest of this section refer to $z_j$-derivatives on the
disk under consideration.

\begin{proposition}[Residue data]\label{prop:residue-data}
For the curve, line bundles, fixed fourth-root isomorphism, and six
vectors $w_j\in H^0(\Gamma,\Phi)^*$ of
Proposition~\ref{prop:curve-data}, one can choose the basis $u$ with
unchanged determinant so that $(u_1)_0$ is an unramified fiber of $f$
disjoint from $Z$.  Set $r=\xi u$ and define
\[
 v_j^*=\iota^*w_j\in W^*,\qquad V=\sum_{j=1}^6v_j^*.
\]
For any mutually disjoint compatible coordinate disks as above there
are nonzero constants $c_j$ and a global pair
\[
 P=(P_1,P_2)\in H^0(C,\theta+Z)^{\oplus2}
\]
with the following properties.  First,
\begin{equation}\label{trace:vector-identities}
 c_jl_j=v_j^*,\qquad
 V^2=90=9\deg L,\qquad (v_j^*)^2+v_j^*\cdot V=1.
\end{equation}
On disk $j$, using coefficient rows in the indicated frames, put
\begin{equation}\label{trace:local-data}
 m_j=\frac{c_j}{z_j},\qquad
 R_j=m_jr,\qquad
 W_j=R_j(0)\wedge R_j'(0),\qquad
 p_j=P-m_jr'.
\end{equation}
The pair $P$ has principal part $R_j(0)/z_j$ in the frame
$(\ell_j^{\mathrm{fr}})^2$ of $\theta$; the rows $R_j,p_j$ are
holomorphic on the disk, and $R_j(0)\ne0$.  Moreover,
\begin{equation}\label{trace:residue-identities}
 W_j=(v_j^*)^2,\qquad
 p_j(0)\wedge R_j(0)=1,\qquad
 \sum_{j=1}^6(W_j-1)=-9\deg L.
\end{equation}
Globally,
\begin{equation}\label{trace:J}
 J=u\wedge P\in H^0(C,\Lambda),\qquad
 \langle[J],[T]\rangle=-V([T])
       \quad\text{for every }T\in H^0(C,\Lambda).
\end{equation}
Changing $P$ by a holomorphic pair of sections of $\theta$ does not
change $[J]$.  The pole prescription is consistent under changes of
compatible coordinates.
\end{proposition}

\begin{proof}
A determinant-one change of basis preserves $\eta_C$ and
$\eta_\Gamma$, and hence all the forms already fixed.  Avoiding the
finitely many branch values and the image of $Z$, we may arrange that
the zero divisor of $u_1$ is the required fiber.  The isometry of
Lemma~\ref{trace:principal-parts} transports the equations for $w_j$
to \eqref{trace:vector-identities}.  It also transports their evaluation
directions.  Since each $l_j$ is nonzero, there is a unique nonzero
$c_j$ with $c_jl_j=v_j^*$.
Locally $R_j=c_ju$, so it is holomorphic and nonzero at the center, and
\eqref{trace:compatible-norm} gives
\[
 R_j(0)\wedge R_j'(0)=c_j^2l_j^2=(v_j^*)^2.
\]

Every prescribed set of simple principal parts of $\theta$ at $Z$
is realizable by a section of $\theta+Z$.  Indeed, the obstruction in
$H^1(C,\theta)$ pairs, by Serre duality, with
$H^0(C,K_C-\theta)=H^0(C,\theta)$ by the sum of residues.
All these test sections vanish at $Z$, by
Proposition~\ref{prop:curve-data}.  Their residue against any simple
principal part is therefore zero.  Apply this separately to the two
entries to obtain $P$ with the stated pole parts.  Since
$r=z_ju$, the row $m_jr'$ has exactly the same simple pole part as $P$;
thus $p_j$ is holomorphic.

The simple pole of $P$ at $q_j$ is proportional to $u(q_j)$.  Its
wedge with $u$ has zero residue coefficient, so $J=u\wedge P$ is
holomorphic there; away from $Z$ it is already holomorphic.  Hence it
is a global section of $\Lambda$.  To compute its class, fix
$T\in H^0(C,\Lambda)$ and consider the global meromorphic canonical
differential
\[
 \nu=\frac{P_1T}{u_1},
 \qquad \theta+(\theta+H)-H=2\theta=K_C.
\]
Its only possible poles lie in $Z$ and in the simple zero fiber of
$u_1$.  At $q_j$, the compatible frames make the canonical product
exactly multiplication of coefficients followed by $dz_j$, and therefore
\[
 \Res_{q_j}\nu=c_jl_j(T).
\]
At a zero $q$ of $u_1$, the section $u_2$ is nonzero and
\[
 J(q)=-u_2(q)P_1(q),\qquad
 \eta_C(q)=-u_2(q)\,du_1(q).
\]
Both expressions have the same minus sign.  In a local parameter,
the residue is consequently
\[
 \Res_q\nu=\left(\frac{JT}{\eta_C}\right)(q).
\]
The zero fiber is unramified, so the sum of these four residues is
the value of $\Tr(JT/\eta_C)$ there.  This trace is a global
holomorphic function on $\PP^1$ by Lemma~\ref{trace:pairing}, hence
is constant.  The residue theorem gives
\[
 \langle[J],[T]\rangle
       =-\sum_jc_jl_j(T)=-V([T]),
\]
which proves \eqref{trace:J} with the stated sign.

On a fixed disk suppress the subscript $j$.  Since $r=zu$,
\[
 P=p+\frac{c}{z}u+cu',\qquad
 cJ=R\wedge p+R\wedge R'.
\]
In particular,
\[
 c_jl_j(J)=R_j(0)\wedge p_j(0)+W_j.
\]
On the other hand, the metric identification in \eqref{trace:J} gives
\[
 c_jl_j(J)=-v_j^*\cdot V=W_j-1.
\]
Thus $R_j(0)\wedge p_j(0)=-1$, or equivalently
$p_j(0)\wedge R_j(0)=1$.  Summing
$W_j-1=-v_j^*\cdot V$ proves the last identity in
\eqref{trace:residue-identities}.  Notice also the useful local expansion
\begin{equation}\label{trace:P-constant}
 P=\frac{R_j(0)}{z_j}
       +p_j(0)+2R_j'(0)+O(z_j).
\end{equation}
If $P$ is changed by a holomorphic pair $S$, the equality
$H^0(C,\theta)=\xi^2H^0(C,2H)$ shows that
$u\wedge S\in\xi^2H^0(C,3H)$; thus $[J]$ is unchanged.

For completeness, let $\zeta=\psi(z)$ be another centered parameter,
and let the corresponding compatible frame of $L$ be $g(z)\ell^{\mathrm{fr}}$.
Compatibility says $g^4=\psi'$.  The associated changes of the other
frames are
\[
 a_{\rm new}=\frac{z}{\psi(z)}a,\qquad
 \mathfrak f_{\rm new}=g\frac{\psi(z)}{z}\mathfrak f,
 \qquad \tau_{\rm new}=g^3\frac{\psi(z)}{z}\tau.
\]
Writing $g_0=g(0)$, it follows that
$l_{\rm new}=g_0^{-7}l$, so the same vector $v^*$ prescribes
$c_{\rm new}=g_0^7c$.  Also
\[
 r_{\rm new}=g^{-1}r,\qquad
 R_{\rm new}(0)=g_0^2R(0).
\]
The coefficient row of the global $\theta$-section transforms as
$P_{\rm new}=g^{-2}P$.  Its coefficient of $\zeta^{-1}$ is
$\psi'(0)g_0^{-2}R(0)=g_0^2R(0)$, exactly the newly prescribed
$R_{\rm new}(0)$.  Thus the same global $P$ realizes the new
prescription.  The row $R_{\rm new}$ is a scalar multiple of $R$,
with scalar $g_0^2$ at the center; differentiating with respect to
$\zeta$ shows $W_{\rm new}=g_0^4W/\psi'(0)=W$.
Finally, subtracting $m_{\rm new}\,d r_{\rm new}/d\zeta$ from
$P_{\rm new}$ shows that
$p_{\rm new}(0)-g_0^{-2}p(0)$ is proportional to $R(0)$.
Hence $p_{\rm new}(0)\wedge R_{\rm new}(0)=p(0)\wedge R(0)$.
All the local conclusions are therefore compatible with the fixed
fourth-root structure.
\end{proof}
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//03-trace.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//04-jets.tex ====
\section{Gluing the vector-field jets}\label{jets:section}

We use the curve and sections of Proposition~\ref{prop:curve-data},
with the normalization supplied by Proposition~\ref{prop:residue-data}.
Thus $L^4\simeq K_C$, $\deg L=10$, and
$r=(r_1,r_2)=\xi(u_1,u_2)$ is a basis of $H^0(C,L)$ with simple common
zero divisor $Z=q_1+\cdots+q_6$.
For rows $X,Y$ put $X\wedge Y=X_1Y_2-X_2Y_1$.
In the compatible coordinates at $q_j$, the preceding proposition gives
\begin{equation}\label{jets:residue-input}
 \begin{gathered}
 m_j=\frac{c_j}{z},\qquad R_j=m_jr,\qquad p=P-m_jr',\qquad
 p(0)\wedge R_j(0)=1,\\
 \sum_{j=1}^6\bigl(W_j-1\bigr)=-9\deg L,
 \qquad W_j=R_j(0)\wedge R_j'(0).
 \end{gathered}
\end{equation}
Here $P$ is a meromorphic row of sections of $L^2$, holomorphic off $Z$,
and $p$ and $R_j$ are holomorphic on the disk in question.
The projective directions of $r'(q_j)$ are not all equal.
Primes always denote differentiation in that disk's base coordinate.
Our task is to choose finitely many neighborhood transitions and compatible
vector-field coefficients whose determinant has the required leading term.
We first solve the exceptional low-order obstruction, then extend the fields
and their determinant together. Proposition~\ref{prop:jets} gives the finite
jet needed for holomorphic lifting.

\subsection{Coordinates, transitions, and obstruction spaces}

On the total space of $L^{-1}$ choose a base coordinate $z$ and a
fiber-linear equation $t$ of the zero section such that $t^4$ corresponds
to $dz$ under $L^4\simeq K_C$.  Two such choices satisfy
\[
 \widetilde z=\psi(z),\qquad \widetilde t=g(z)t,
 \qquad g^4=\psi'.
\]
Put $E_t=t\partial_t$.  The meromorphic two-form
\begin{equation}\label{jets:volume}
 \sigma=t^{-4}\frac{dt}{t}\wedge dz=t^{-5}dt\wedge dz
\end{equation}
is invariant under these split changes of coordinates.
We seek rows of coefficients $A_k,B_k$ defining
\begin{equation}\label{jets:fields}
 \begin{gathered}
 v_i=t\bigl(A_i(t,z)E_t+B_i(t,z)\partial_z\bigr),\\
 A=\sum_{k\geq0}A_kt^k,\quad B=\sum_{k\geq0}B_kt^k,
 \qquad A_0=r,\quad B_0=0.
 \end{gathered}
\end{equation}
The subscripts $k$ label orders; subscripts $i,j\in\{1,2\}$ label
entries of a row.  In particular,
\begin{equation}\label{jets:determinant}
 \sigma(v_1,v_2)=t^{-2}(A\wedge B),\qquad
 v_1\wedge v_2=t^2(A\wedge B)E_t\wedge\partial_z.
\end{equation}

For a term $t^{k+1}(A_kE_t+B_k\partial_z)$, a split coordinate change
gives, with $a=(\log g)'$,
\begin{equation}\label{jets:split-coefficients}
 \widetilde A_k=g^{-(k+1)}(A_k+aB_k),\qquad
 \widetilde B_k=g^{-(k+1)}\psi' B_k=g^{3-k}B_k.
\end{equation}
Consequently the horizontal coefficient has type $L^{k-3}$, and the
difference of two vertical lifts of a fixed horizontal coefficient has
type $L^{k+1}$.  More precisely, the sheaf of homogeneous terms of this
order is an extension
\begin{equation}\label{jets:homogeneous-extension}
 0\longrightarrow L^{k+1}\longrightarrow\mathcal E_k
 \longrightarrow L^{k-3}\longrightarrow0.
\end{equation}
This also specifies how rows written in local coordinates are interpreted
on a chart without a global coordinate or frame.

Take the cover consisting of $U_0=C\setminus Z$ and six mutually
disjoint disks $U_j$ about the $q_j$. The positive-degree divisor $Z$
has a very ample multiple; the corresponding projective embedding
realizes $U_0$ as a closed algebraic subvariety of the affine complement
of a hyperplane. Its analytification is therefore Stein. The disks and
annular intersections are Stein as well.  Thus horizontal sections admit holomorphic vertical lifts
on each chart, by \eqref{jets:homogeneous-extension} and Cartan's
Theorem~B~\cite[\S4, Theorem~B]{Cartan1952}.  Cousin problems on this cover compute the indicated first
cohomology groups.  Equivalently, one may shrink to a cover with annular
intersections.  For an overlap differential we use the contour residue
$\frac1{2\pi i}\int\omega$ around the missing point, oriented positively
in the disk coordinate.  The Serre-duality obstruction to a splitting
is the sum of these residues after multiplication by a dual section.
This formulation allows overlap functions holomorphic on annuli; they
need not be meromorphic at the omitted centers.

We modify the disk-to-outside identifications by flows of the vector
fields
\begin{equation}\label{jets:generator}
 D=t^k(bE_t+h\partial_z),\qquad k>0,
\end{equation}
where $b,h$ are holomorphic on the overlap in the disk coordinate.
Use the time-one flow of $-D$, so that its action on vector fields is
$\exp(\operatorname{ad}D)$.  Successive modifications are applied to
rows already expressed on the outside; all formulas below are relative
to the original split comparison.  Every generator vanishes on the
zero section and changes its equation only by $t\mapsto t+O(t^2)$,
so the zero section and its conormal bundle are preserved.

\begin{lemma}\label{jets:bracket}
For $D$ as in \eqref{jets:generator} and
$V=t^{\ell+1}(AE_t+B\partial_z)$ one has
\begin{align}
 [D,V]=t^{k+\ell+1}\bigl(&((\ell+1-k)bA+hA'-Bb')E_t\notag\\
 &+((\ell+1)bB+hB'-khA-Bh')\partial_z\bigr).
 \label{jets:general-bracket}
\end{align}
Moreover,
\begin{equation}\label{jets:divergence}
 \mathcal L_D\sigma=t^k\bigl((k-4)b+h'\bigr)\sigma.
\end{equation}
In particular, for $k\ne4$ the choice $b=h'/(4-k)$ preserves $\sigma$
exactly.  The first effect of an order-$k$ modification on
\eqref{jets:fields} is
\begin{equation}\label{jets:leading-action}
 \Delta A_k=(1-k)br+hr',\qquad \Delta B_k=-khr.
\end{equation}
\end{lemma}

\begin{proof}
Use $E_t(t^a)=at^a$ and $[E_t,\partial_z]=0$ in the two components of
the bracket.  For the divergence, the coefficient of $dt\wedge dz$ in
$\sigma$ is $t^{-5}$, so
\[
 \operatorname{div}_{\sigma}D
 =t^5\left(\partial_t(t^{k-4}b)
              +\partial_z(t^{k-5}h)\right)
 =t^k((k-4)b+h').
\]
Setting $\ell=0$, $A=r$, $B=0$ gives the last formula.
\end{proof}

All calculations can initially be made with finite Taylor series.
They nevertheless prescribe actual transition germs: on a compact
subannulus every $D$ in \eqref{jets:generator} is holomorphic near its
zero section and vanishes there, so its time-one flow and inverse exist
after shrinking in the transverse direction.  A finite composition has
the computed Taylor series.  There are no intersections between
different disk charts, hence no additional triple-overlap equations.
The passage from these germs to a Hausdorff surface neighborhood will
be made in the next section.

\subsection{The first two orders}

At order one use $h=m=m_j$, $b=m'/3$ on the $j$th overlap, and choose
\begin{equation}\label{jets:first-order}
\begin{array}{c|cc}
 &A_1&B_1\\ \hline
 U_j&p&mr\\
 U_0&P&0.
\end{array}
\end{equation}
By \eqref{jets:leading-action}, the first transition adds
$(mr',-mr)$ to the inside row, giving exactly $(P,0)$.
The degree-one coefficient of $A\wedge B$ is zero on both charts.

At order two write $U=B_2$.  To give the determinant its required
leading coefficient, solve on each chart
\begin{equation}\label{jets:second-horizontal}
 r\wedge U=1-A_1\wedge B_1.
\end{equation}
On a disk write $r=z\rho$, with $\rho$ nowhere zero as a row after
shrinking.  By \eqref{jets:residue-input}, the right side of
\eqref{jets:second-horizontal} vanishes at $z=0$, so division by $z$
reduces it to a surjective wedge map with $\rho$.
On $U_0$ the row $r$ has no common zero, and the corresponding
surjection of coherent sheaves is surjective on sections by
Steinness.  Thus the required holomorphic horizontal rows exist.
After the first transition, their horizontal difference on an overlap
wedges to zero with $r$: the determinant target is compatible because
that transition preserves $\sigma$.
The difference is therefore a unique holomorphic multiple of $r$.
An order-two transition with $b=h'/2$ removes it, since its horizontal
effect is $-2hr$.  Keep the chosen outside row $U$ fixed throughout this
matching.

It remains to match the vertical lifts $A_2$.  The obstruction belongs
to $H^1(C,L^3)^2$.  Its four Serre-duality tests use $r_1,r_2$, a basis
of $H^0(C,K_C\otimes L^{-3})=H^0(C,L)$, against the two entries of
the vertical difference.  The following calculation evaluates all four.

\begin{lemma}\label{jets:second-obstruction}
The order-two vertical obstruction vanishes.
\end{lemma}

\begin{proof}
Write $D^{(2)}=A_2$; this notation denotes the vertical row, not a
transition generator.  For $i,j\in\{1,2\}$ set
\begin{equation}\label{jets:psi}
 \Psi_{ij}=\left(r_iD_j^{(2)}+\frac14r_i'U_j
                          +\frac34r_j'U_i\right)dz.
\end{equation}
Once the horizontal rows have matched, the difference of these
expressions for the transported inside lift and the outside lift is
exactly $r_i$ times the $j$th vertical difference, multiplied by $dz$.
The derivative terms therefore change neither the obstruction nor its
four tests.

First compute the total for the outside lift.  Put
$\epsilon_{12}=1$, $\epsilon_{21}=-1$, and
$\epsilon_{11}=\epsilon_{22}=0$.
Under a split change with $a=(\log g)'$, formula
\eqref{jets:split-coefficients} gives
\[
 \widetilde r_i=g^{-1}r_i,\qquad
 \partial_{\widetilde z}\widetilde r_i=g^{-5}(r_i'-ar_i),\qquad
 \widetilde U_i=gU_i,\qquad
 \widetilde D_i^{(2)}=g^{-3}(D_i^{(2)}+aU_i).
\]
Substituting in \eqref{jets:psi}, and expressing both differentials in
the old coordinate, yields
\begin{equation}\label{jets:psi-transformation}
 \widetilde\Psi_{ij}-\Psi_{ij}
 =\frac34(r_iU_j-r_jU_i)d\log g
 =\frac34\epsilon_{ij}\,d\log g
 \quad\hbox{on }U_0.
\end{equation}
The last equality uses $r\wedge U=1$ on the outside.
Use the section $r_1=\xi u_1$ of $L$.  Its coefficient transforms by
$\widetilde r_1=g^{-1}r_1$, so
$\Psi_{ij}+\frac34\epsilon_{ij}d\log r_1$ is an intrinsic differential
on $U_0$.  There it has precisely the four possible simple poles at
the reduced zero fiber of $u_1$, each of residue
$\frac34\epsilon_{ij}$.  The residue theorem on the complement of
small disks therefore gives total contour residue $-3\epsilon_{ij}$
around $Z$ for this intrinsic differential.  At each of the six
points of $Z$, the section $r_1$ has a simple zero.  Subtracting these
six logarithmic residues yields
\begin{equation}\label{jets:outside-residue}
 \sum_{q\in Z}\Res_q\Psi_{ij}^{\mathrm{out}}
 =-\frac34\epsilon_{ij}\deg L.
\end{equation}
Only contour integrals at $Z$ were used, so this argument does not
assume meromorphic extension of the outside lifts.

Next compute the transported inside total.  Before transport,
$\Psi_{ij}$ is holomorphic on the disk.  The order-two transition
changes its rows by
\[
 \delta D^{(2)}=hr'-\frac12h'r,\qquad \delta U=-2hr.
\]
In \eqref{jets:psi} this is the exact differential
\begin{equation}\label{jets:psi-exact}
 \delta\Psi_{ij}=-\frac12d(hr_ir_j),
\end{equation}
so it has zero contour residue.
It suffices to calculate the effect of the order-one transition,
including its second exponential term.

The bracket of $t(m'E_t/3+m\partial_z)$ with the inside order-one
term $t^2(pE_t+mr\partial_z)$ contributes
\[
 \left(\frac13m'p+mp'-\frac13mm''r,
              \frac23mm'r+m^2r'-mp\right)
\]
to $(A_2,U)$.  Half of its second bracket with the leading term
$trE_t$ contributes
\[
 \left(\frac23mm'r'+\frac12m^2r''+\frac16mm''r,
              -\frac13mm'r-m^2r'\right).
\]
Thus the full effect is
\begin{align}
 \Delta A_2&=\frac13m'p+mp'+\frac23mm'r'
                         +\frac12m^2r''-\frac16mm''r,
 \notag\\
 \Delta U&=-mp+\frac13mm'r.
 \label{jets:second-expansion}
\end{align}
The $m^2r'$ horizontal terms cancel in this sum.

Fix one disk and abbreviate $R=mr$, $m=c/z$, so $r=zR/c$.
Substitution in \eqref{jets:second-expansion} gives
\[
 \begin{aligned}
 \Delta A_2&=-\frac{cR}{z^3}+\frac{cR'}{3z^2}
             +\frac{cR''}{2z}-\frac{cp}{3z^2}+\frac{cp'}z,
 \\
 \Delta U&=-\frac{cp}z-\frac{cR}{3z^2}.
 \end{aligned}
\]
It follows, with every coefficient on the right evaluated at $0$, that
\begin{align}
 F_{ij}:=\Res_0(r_i\Delta A_{2,j}\,dz)
   &=-\frac13R_ip_j-\frac23R_iR_j'-R_i'R_j,\notag\\
 H_{ij}:=\Res_0(r_i'\Delta U_j\,dz)
   &=-R_ip_j-\frac13R_iR_j'-\frac23R_i'R_j.
 \label{jets:laurent-residues}
\end{align}
For example, the double pole $-R_iR_j/z^2$ in the first expression
contributes $-R_i'R_j-R_iR_j'$ to its residue; this accounts for both
derivative terms.  The complete residue of the change in
\eqref{jets:psi} is consequently
\begin{equation}\label{jets:full-residue-matrix}
 \begin{aligned}
 T_{ij}&=F_{ij}+\frac14H_{ij}+\frac34H_{ji}\\
 &=-\frac7{12}R_ip_j-\frac9{12}R_jp_i
                -\frac54R_iR_j'-\frac{17}{12}R_i'R_j.
 \end{aligned}
\end{equation}

Decompose this matrix into half its sum and half its difference with
its transpose.  Directly from \eqref{jets:full-residue-matrix},
\begin{align}
 T_{ij}^{\mathrm{sym}}
 &=-\frac23\bigl(R_ip_j+R_jp_i
                         +2R_iR_j'+2R_i'R_j\bigr),\notag\\
 T_{ij}^{\mathrm{alt}}
 &=\frac1{12}\epsilon_{ij}
                 \bigl((R\wedge p)(0)+(R\wedge R')(0)\bigr).
 \label{jets:symmetric-alternating}
\end{align}
To identify the symmetric term intrinsically, observe that
\[
 P=p+mr'=\frac{R(0)}z+p(0)+2R'(0)+O(z).
\]
Since $P_iP_j$ is a meromorphic section of $L^4=K_C$, its local
expression is $P_iP_j\,dz$, and
\[
 \Res_0(P_iP_j\,dz)
 =R_ip_j+R_jp_i+2R_iR_j'+2R_i'R_j.
\]
Thus the sum of every symmetric entry in
\eqref{jets:symmetric-alternating} is zero, by the residue theorem
on $C$.  For the alternating entry,
\eqref{jets:residue-input} gives $R(0)\wedge p(0)=-1$ and therefore
\[
 \sum_{q\in Z}T_{ij}^{\mathrm{alt}}
 =\frac1{12}\epsilon_{ij}\sum_{\nu=1}^6(W_\nu-1)
 =-\frac34\epsilon_{ij}\deg L.
\]
In particular, the sums of the four entries $(11),(12),(21),(22)$
of the transported inside matrix are respectively
\[
 0,\qquad -\frac34\deg L,\qquad
 \frac34\deg L,\qquad 0.
\]
They coincide with the four outside totals in
\eqref{jets:outside-residue}.  Hence every Serre-duality pairing of the
vertical difference vanishes.  The corresponding class in
$H^1(C,L^3)^2$ is zero, so holomorphic vertical corrections on the
charts complete the order-two matching.
\end{proof}

\subsection{Higher orders and determinant targets}

From order three onward, each step extends both the fields and the
determinant they are required to have.  Write the determinant equation as
\begin{equation}\label{jets:target-equation}
 A\wedge B=t^2G,\qquad G=1+G_1t+G_2t^2+\cdots.
\end{equation}
The field equation in degree $k$ uses $G_{k-2}$.  To compare the next
horizontal rows, this target coefficient must already be compatible
on the overlaps.  We therefore choose each target coefficient before
imposing its field equation, and keep track of these two extensions
separately.

The target is the local bivector $G\sigma^{-1}$.  On a realization $Y$,
with $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the relevant line bundle is
\[
 \mathcal M=\cI^4\det\cF=\cI^5\det\cT_Y,
 \qquad \mathcal M|_C=L^4\otimes\cT_C\simeq\OO_C.
\]
Its local frame is $\sigma^{-1}$, and its degree-$k$ associated graded
is $L^k$.  Only its leading trivialization on $C$ is fixed initially;
the successive targets extend that section through finite orders.
After transitions cease to preserve $\sigma$, $G$ is therefore not
an invariant scalar function.  To be explicit, if
$q=\operatorname{div}_{\sigma}D$, then
\begin{equation}\label{jets:target-transport}
 [D,G\sigma^{-1}]=(D(G)-qG)\sigma^{-1},\qquad
 G\longmapsto\exp(D-q)G.
\end{equation}
Here $D-q$ denotes the differential operator $f\mapsto D(f)-qf$.
An order-$k$ transition changes a target with constant term one first
in degree $k$ of $G$, with contribution
$-((k-4)b+h')$.  Once lower coefficients are compatible, a leading
degree-$k$ discrepancy is a cocycle in $L^k$.

The factor $t^2$ in \eqref{jets:target-equation} and the equality
$B_0=0$ determine the order of the choices.  For the field changes
at step $k$, the first determinant equations they can affect are
\[
\begin{array}{c|c|c}
 \text{changed field coefficient}
 &\text{degree in }A\wedge B&\text{target coefficient}\\ \hline
 \text{global addition to }A_{k-1}&k&G_{k-2}\\
 B_k&k&G_{k-2}\\
 A_k&k+1&G_{k-1}.
\end{array}
\]
The target coefficients in the last column remain fixed during these
changes.  In particular, we can still adjust $A_{k-1}$ to solve the
degree-$k$ equation without changing any determinant equation imposed
at an earlier step.  A new target coefficient $G_k$ will be chosen
only after the fields match through index $k$; its field equation
occurs in degree $k+2$.

We will maintain the following invariant after step $n\geq2$:
the field rows through index $n$ match; the determinant equations
\eqref{jets:target-equation} are imposed through degree $n$ in
$A\wedge B$; and the coefficients $G_0=1,G_1,\ldots,G_n$ are chosen
and compatible through degree $n$ under the transition maps.
Thus target compatibility is known two determinant degrees beyond
the equations already imposed: the equations involving $G_{n-1}$
and $G_n$ are still to be solved.
The preceding construction establishes this invariant for $n=2$,
with $G=1$, since both transitions preserve $\sigma$ exactly.

Suppose now that the invariant holds through step $k-1$, where
$k\geq3$.  The coefficient needed in the current determinant equation,
$G_{k-2}$, has already been chosen.  On each chart the horizontal row
must solve
\begin{equation}\label{jets:horizontal-induction}
 r\wedge B_k=G_{k-2}
       -\sum_{a=1}^{k-1}A_a\wedge B_{k-a}.
\end{equation}
This is an equation in $L^{k-2}$.  Indeed, under
\eqref{jets:split-coefficients}, the only additional terms in the
transformation of the sum on the right are multiples of
\[
 \sum_{a=1}^{k-1}B_a\wedge B_{k-a}=0;
\]
the terms cancel in pairs and any middle self-wedge vanishes.

\paragraph{Making the right side divisible at $Z$.}
We may add any row $a\in H^0(C,L^k)^2$ to the previously chosen
vertical coefficient $A_{k-1}$.  Differences of vertical lifts have
this tensor type by \eqref{jets:homogeneous-extension}.
Such a change preserves matching through index $k-1$: every
previous nonsplit transition has positive order, so its action on the
new term $t^kaE_t$ only changes indices at least $k$.
It also preserves all determinant equations already imposed, because
$B_0=0$; its first determinant contribution is $a\wedge B_1$ in
degree $k$.  In particular, it need not preserve the as yet unimposed
equation whose target is $G_{k-2}$.

At each $q_j$ the row $B_1(q_j)=R_j(0)$ is nonzero.
The interpolation in Proposition~\ref{prop:curve-data}(4) gives the
surjection
\[
 H^0(C,L^k)\longrightarrow H^0(Z,L^k|_Z)\qquad(k\geq3).
\]
It permits arbitrary values of both entries of $a$ at all six points.
The scalar maps
$a(q_j)\mapsto a(q_j)\wedge B_1(q_j)$ are surjective, so this
freedom makes the right side of \eqref{jets:horizontal-induction}
vanish at every $q_j$.
Since $r=z\rho$ has only a simple common zero, no derivative
condition is required.  As at order two, division by $z$ on each
disk and Steinness on $U_0$ now give holomorphic solutions $B_k$.

Although this global addition preserves all lower-order matching,
the previous transitions can act on it in the degree currently being solved.
The first transition contributes, at row index $k$,
\begin{equation}\label{jets:old-transition-effect}
 \delta A_k=\frac{k-1}{3}m'a+ma',\qquad
 \delta B_k=-ma.
\end{equation}
All other previous generators, and repeated brackets, affect later
indices.  The displayed horizontal contribution satisfies
$r\wedge(-ma)=a\wedge(mr)$, exactly the corresponding determinant
change on the disk.  We include these terms in the horizontal and
vertical comparisons below.

\paragraph{Matching the horizontal rows.}
Choose any holomorphic vertical lifts of the newly selected $B_k$.
After transport by the previous transitions, the field rows already
agree at every lower index.  The difference of their degree-$k$
determinants is therefore $r\wedge\delta B_k$, since
$\delta A_k\wedge B_0=0$.  Target compatibility through degree
$k-2$ in $G$ makes this difference zero.  Thus the horizontal
discrepancy is a holomorphic multiple of $r$ on each overlap, and
\eqref{jets:leading-action} removes it by a choice of $h$ in an
order-$k$ transition.  Use $b=h'$ when $k=3$, and $b=0$ when
$k\geq4$.
The new transition first affects $G$ in degree $k$, so cannot alter
the target comparison just used in degree $k-2$.

\paragraph{Matching the vertical rows.}
After horizontal matching, the remaining class is in
$H^1(C,L^{k+1})^2$.
For $k\geq4$ it vanishes, since
\begin{equation}\label{jets:vertical-vanishing}
 H^1(C,L^{k+1})\simeq H^0(C,L^{3-k})^*=0.
\end{equation}
Holomorphic vertical corrections on the charts therefore solve the
matching.  They do not change the current determinant equation,
because $B_0=0$.

At $k=3$ the obstruction is $H^1(C,K_C)^2$, tested by the constants
in the two entries.  There is additional overlap freedom to kill it.
On the $j$th overlap replace $h$ by $h+\lambda_j/z$ and compensate
on its disk by
\begin{equation}\label{jets:third-compensation}
 B_3\longmapsto B_3+\frac{3\lambda_jr}{z}.
\end{equation}
This is holomorphic and its wedge with $r$ is zero.  It preserves
both the current determinant equation and horizontal matching:
the transition adds the opposite horizontal row $-3\lambda_jr/z$.
Keep the vertical lift $A_3$ unchanged in the chosen disk coordinate.
This prescribes a holomorphic section of $\mathcal E_3$ on that disk;
in another split coordinate its vertical component acquires the term
$a\,\delta B_3$ dictated by \eqref{jets:split-coefficients}.
Changing a degree-three input row affects its transport under old
positive-order transitions only at later indices.
The change in the transported vertical difference at index three is
therefore exactly
\begin{equation}\label{jets:third-residue}
 \lambda_j\left(\frac{r'}z+\frac{2r}{z^2}\right),\qquad
 \Res_0\left[\lambda_j\left(\frac{r'}z+\frac{2r}{z^2}\right)dz\right]
       =3\lambda_jr'(0).
\end{equation}
The two entries here are coefficients of canonical differentials,
since $L^4=K_C$.  The six rows $r'(q_j)$ span $\CC^2$: their
projective directions are those of the pencil values $u(q_j)$, which
are not all the same.  Hence the map
$(\lambda_1,\ldots,\lambda_6)\mapsto\sum_j3\lambda_jr'(q_j)$
has rank two and can cancel both obstruction entries.
The resulting vertical class splits, completing order three.

The correction \eqref{jets:third-compensation} can change the
degree-four coefficient of $A\wedge B$.  This causes no conflict:
$G_2=0$ has been selected, but its determinant equation is first
imposed at step four.  The degree-three equation already imposed is
unchanged.  Moreover the modified order-three transition still has
$b=h'$ and so preserves $\sigma$ exactly, leaving the transformation
law of the chosen targets intact.

\paragraph{Extending the determinant target.}
It remains to choose $G_k$ compatibly after the order-$k$ transition.
For $k=3$ all transitions preserve $\sigma$, so we retain $G=1$.
At $k=4$ the earlier transitions still preserve it exactly; thus the
first target discrepancy is $-h'$ in degree four, by
\eqref{jets:target-transport} with $b=0$.
The horizontal coefficient of an order-four generator has type
$L^4\otimes\cT_C=\OO_C$.  More explicitly, its split
transformation is $\widetilde h=g^{4-k}h$, which equals $h$ at
$k=4$.  Consequently $h'dz=dh$ is an intrinsic exact differential
on the overlap.  Each such differential has zero contour residue.
The sole obstruction to splitting this $L^4=K_C$ cocycle is its
pairing with the constant section of $\OO_C$, so it vanishes.
There are holomorphic chart coefficients $G_4$ accounting for the
discrepancy, with $G_1=G_2=G_3=0$.
Transport by a previous positive-order modification changes a
degree-four term only in degrees at least five. Equivalently, conjugating
the order-four generator by the earlier transitions leaves its degree-four
part unchanged. Thus the scalar transformation law of $h$, and the residue
calculation using $dh$, apply intrinsically at this order.

For $k>4$, once the lower target coefficients are compatible, the
new leading discrepancy lies in $L^k$, and
\begin{equation}\label{jets:target-vanishing}
 H^1(C,L^k)\simeq H^0(C,L^{4-k})^*=0.
\end{equation}
Thus it always splits and defines $G_k$.
In every case this target extension leaves all equations already
imposed unchanged, since $G_k$ is used only in degree $k+2$ of
$A\wedge B$.  This completes the induction invariant.

We now choose where to stop.  A row of index $k$ contributes degree
$k+1$ in the logarithmic frame $(E_t,\partial_z)$, so matching through
index seven prescribes a section modulo $\cI^9\cF$.  This is the
threshold needed in Section~\ref{analytic:section}: the successive
quotients of the unprescribed tail are extensions of $L^{n-4}$ by
$L^n$, and both line bundles have vanishing first cohomology for
$n\geq9$.  Lemma~\ref{analytic:actual-lifting} turns these graded
vanishings into holomorphic lifting on a realized neighborhood.

\begin{proposition}[Finite compatible jets]\label{prop:jets}
For the data of Propositions~\ref{prop:curve-data} and
\ref{prop:residue-data}, there are finitely many holomorphic
disk-to-outside transition germs, each a composition of flows
\eqref{jets:generator} of orders $1,\ldots,7$, together with
holomorphic local rows $A_k,B_k$ for $0\leq k\leq7$, such that:
the transitions fix $C$ and preserve its conormal bundle $L$;
the vector fields \eqref{jets:fields} match through row index seven;
and $A\wedge B=t^2(1+O(t))$ on every chart.

On any surface neighborhood $Y$ realizing these transition germs,
write $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$.
The rows give two sections of $\cF/\cI^9\cF$, lying in
$\cI\cF/\cI^9\cF$.  Their determinant has leading term
$t^4E_t\wedge\partial_z$, equivalently
$t^5\partial_t\wedge\partial_z$, with coefficient one.
\end{proposition}

\begin{proof}
Perform the first two steps and then the induction through $k=7$.
Only finitely many flows occur, and each is an actual holomorphic
germ on an annular neighborhood of the zero section, as explained
after Lemma~\ref{jets:bracket}.  Terms with row indices $0,\ldots,7$
have logarithmic-frame coefficients of degrees $1,\ldots,8$.
Matching those terms is precisely matching modulo $\cI^9\cF$.
Finally \eqref{jets:determinant} and the degree-two determinant
equation give
\[
 v_1\wedge v_2=t^4(1+O(t))E_t\wedge\partial_z
             =t^5(1+O(t))\partial_t\wedge\partial_z.
\]
This statement uses only finite compatible jets; analytic realization
and the lifting of these quotient sections are addressed next.
\end{proof}
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//04-jets.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//05-analytic.tex ====
\section{Analytic realization and contraction}
\label{analytic:section}

We now realize the finite gluing data of Proposition~\ref{prop:jets} by
an actual complex surface.  The two prescribed vector fields will then
be lifted from their finite jets by a cohomology vanishing argument.
The realization uses only the finite list of genuine holomorphic flow
maps already constructed. The formal-functions theorem used afterwards
computes a cohomology group on that realized neighborhood.
Throughout this section, powers of an ideal denote ordinary ideal
powers, and cohomology is analytic sheaf cohomology. We use coherence
of analytic structure and ideal sheaves as in
\cite[Chapter~II, Theorems~3.19 and~4.29]{Demailly2012}.

\subsection{A Hausdorff neighborhood from the finite overlap maps}

\begin{lemma}\label{analytic:realization}
The overlap maps of Proposition~\ref{prop:jets} define, after shrinking
their domains, a smooth Hausdorff complex surface neighborhood $Y$ of
$C$.  Its conormal bundle is $L$, and hence
\[
 C^2=-10.
\]
Writing $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the prescribed rows
define two global sections
\[
 \bar v_1,\bar v_2\in H^0(Y,\cF/\cI^9\cF).
\]
In divisor coordinates $(t,z)$, their local representatives have
logarithmic determinant $t^4(1+O(t))$ and ordinary determinant
$t^5(1+O(t))$.
\end{lemma}

\begin{proof}
Let $N=L^{-1}$ be the split normal bundle.  All changes to its
disk-to-outside identifications are finite compositions of flows of
holomorphic vector fields
\[
 D=t^k\bigl(b(z)t\partial_t+h(z)\partial_z\bigr),\qquad k\geq1,
\]
on punctured disk neighborhoods.  On a compact annulus in any such
overlap, these vector fields vanish along the zero section.
The local existence theorem for holomorphic differential equations,
applied uniformly on that compact annulus, therefore gives the
required time-one maps and their inverses after shrinking the fiber
radius.  There are only finitely many flows.  Shrinking successively
gives their compositions on neighborhoods of the same compact
annular bands.  Denote the resulting inside-to-outside maps by
$\Phi_j$, $1\leq j\leq6$.

Here is a quotient construction that also verifies the separation
property.  Choose mutually disjoint coordinate disks $D_j$ about the
six points, and in each choose radii
\[
 0<a_j<a'_j<b'_j<b_j
\]
so that $\Phi_j$ and its inverse are defined on neighborhoods of the
closed annulus $a_j\leq |z_j|\leq b_j$ in their respective split
bundles.  Their domains may be taken somewhat wider than these closed
annuli.  Retain a small open neighborhood $V_j$ of the zero section
over $|z_j|<b'_j$, and a small open neighborhood $V_0$ in $N$ over
\[
 C\setminus\bigcup_{j=1}^6\{|z_j|\leq a'_j\}.
\]
Choose the transverse thicknesses small enough that the maps are
defined throughout the retained parts of the closed annuli and
\begin{equation}\label{analytic:collar-bound}
 \bigl|z_j(\Phi_j(p))-z_j(p)\bigr|
 <\frac{a'_j-a_j}{2}
\end{equation}
there.  This is possible because each map restricts to the identity
on the zero section.  We also make the images of the retained annuli
lie over their respective disjoint disks $D_j$.

Set
\[
 A_j=\{p\in V_j:a_j<|z_j(p)|<b_j,\ \Phi_j(p)\in V_0\},
\]
and identify $p\in A_j$ with $\Phi_j(p)\in V_0$.  These are
biholomorphic identifications between open subsets.  A point of
$V_0$ is in at most one of their images, since those images lie over
disjoint disks.  Each equivalence class in
$V_0\sqcup V_1\sqcup\cdots\sqcup V_6$ thus has at most two elements.
In particular the equivalence relation consists exactly of the
diagonal, these six graphs, and their inverses; no further
identifications or triple-overlap equations occur.

Each graph is closed relative to $V_j\times V_0$.  Indeed, suppose
$p_\nu\in A_j$, $p_\nu\to p\in V_j$, and
$\Phi_j(p_\nu)\to q\in V_0$.  The source base radii have limit at
least $a_j$ and less than $b'_j<b_j$.  The flow is defined across the
closed annulus, so continuity gives $q=\Phi_j(p)$.  If the limiting
source radius were $a_j$, inequality~\eqref{analytic:collar-bound}
would give $|z_j(q)|<a'_j$, contrary to $q\in V_0$.  Thus
$a_j<|z_j(p)|<b_j$ and $p\in A_j$, as required.  Possible fiber
boundaries cause no difficulty because both limiting points are
assumed to belong to the retained open pieces.

The equivalence relation is therefore closed.  The quotient map is
open, since it glues open sets by biholomorphisms.  An open quotient
map with closed equivalence relation has Hausdorff target: two
inequivalent representatives have product neighborhoods disjoint
from the relation, and the images of these neighborhoods are
disjoint open sets.  Moreover, the quotient map restricts on every
$V_i$ to an injective open map.  The complex structures on these
pieces consequently give the quotient a smooth complex surface
structure.  The images of their zero sections give the original
compact curve $C$ with its original complex structure.

Every modifying flow fixes $C$ and induces the identity on its
conormal bundle.  In the split identification its coordinate
expansion has $t'=t+O(t^2)$, although its base coordinate may change
by $O(t)$.  Composing with the original split changes therefore
leaves $\cI/\cI^2\simeq L$.  This proves $C^2=-\deg L=-10$.

In the logarithmic basis $(t\partial_t,\partial_z)$ the prescribed
fields have coefficients $tA_i,tB_i$.  Rows with indices $0$ through
$7$ prescribe these coefficients through degree $8$ in $t$.
The matching in Proposition~\ref{prop:jets} is therefore precisely
matching modulo $\cI^9\cF$ on the actual overlap maps.  It gives the
claimed quotient sections.  Their logarithmic determinant is
\[
 t^2(A\wedge B)=t^4(1+O(t)).
\]
Replacing $t\partial_t$ by $\partial_t$ multiplies this determinant
by $t$, giving the stated ordinary determinant.
\end{proof}



\subsection{Vanishing and holomorphic lifting}

The finite jets now live on a Hausdorff surface. To lift them to
holomorphic fields, we prove that the first cohomology of
$\cI^9\cF$ vanishes after shrinking about $C$.
The logarithmic tangent sheaf $\cF$ is locally free of rank two.
Restriction to $C$ gives the exact sequence
\begin{equation}\label{analytic:log-sequence}
 0\longrightarrow\OO_C\longrightarrow\cF|_C
 \longrightarrow\cT_C\longrightarrow0.
\end{equation}
Here the first map sends $1$ to the class of the normal Euler field
$t\partial_t$, and the second is restriction of a tangent vector
field to $C$.  The Euler class is unchanged when the local equation
of $C$ is multiplied by a unit.  The sequence is thus intrinsic; a
splitting is unnecessary.

\begin{lemma}\label{analytic:graded-vanishing}
For every $n\geq9$,
\[
 H^1(C,\cF|_C\otimes L^n)=0.
\]
Consequently, for $\mathcal E=\cI^9\cF$ and every $r\geq1$,
\[
 H^1(Y,\mathcal E/\cI^r\mathcal E)=0.
\]
\end{lemma}

\begin{proof}
Since $K_C\simeq L^4$, tensoring
\eqref{analytic:log-sequence} by $L^n$ gives
\[
 0\longrightarrow L^n\longrightarrow\cF|_C\otimes L^n
 \longrightarrow L^{n-4}\longrightarrow0.
\]
Serre duality identifies the duals of the first cohomology groups of
the outer terms with
\[
 H^0(C,L^{4-n})\quad\hbox{and}\quad H^0(C,L^{8-n}),
\]
respectively.  Both vanish for $n\geq9$, as $\deg L>0$.
The first assertion follows from the cohomology sequence.

Because $C$ is a Cartier divisor and $\cF$ is locally free,
\[
 \cI^n\cF/\cI^{n+1}\cF\simeq\cF|_C\otimes L^n.
\]
The quotient
$\mathcal E/\cI^r\mathcal E=\cI^9\cF/\cI^{9+r}\cF$
has a finite filtration with precisely these graded terms for
$9\leq n<9+r$.  Their first cohomology vanishes by the first
assertion.  Successive cohomology sequences give the second
assertion; the quotient is supported on the compact curve $C$.
\end{proof}

\begin{lemma}\label{analytic:actual-lifting}
After shrinking $Y$ about $C$, the sections $\bar v_1,\bar v_2$ lift
to holomorphic sections $v_1,v_2$ of $\cF$.  They form a frame for
$\cT_{Y\setminus C}$.
\end{lemma}

\begin{proof}
The intersection matrix of $C$ is the negative definite matrix
$(-10)$.  Grauert's contraction criterion therefore contracts $C$
to a point $x$ of a normal complex surface:
\[
 \pi:Y\longrightarrow X,\qquad \pi^{-1}(x)=C,
 \qquad Y\setminus C\simeq X\setminus\{x\},
\]
where the fiber equality is set-theoretic, and
$\pi_*\OO_Y=\OO_X$.
We use the criterion in \cite[\S4.8(e), pp.~366--367]{Grauert1962}.
The contraction is proper and is a biholomorphism off the
exceptional curve.  It is the holomorphic reduction in that theorem,
whose target is normal when the source is normal; these properties
are part of the reduction and contraction
statements in \cite[\S2, p.~337 and Definition~3,
pp.~339--340]{Grauert1962}.  We choose a sufficiently small Stein
representative $X$ about $x$ and replace $Y$ by its inverse image.
In particular $X\setminus\{x\}$ is smooth.

We prove $R^1\pi_*\mathcal E=0$ for
$\mathcal E=\cI^9\cF$, keeping track of the two ideal filtrations.
Let $\mathfrak m_x$ be the ideal sheaf of $x$ in $X$ and put
$\mathcal J=\mathfrak m_x\OO_Y$.  Both $\mathcal J$ and $\cI$
vanish exactly on $C$, and $\cI$ is radical.  The analytic
Nullstellensatz~\cite[Chapter~II, Theorem~4.22]{Demailly2012}
and Noetherianity give an inclusion
$\cI^a\subseteq\mathcal J$ locally at each point of $C$ for some
positive integer $a$.  A finite cover of the compact curve gives a
single exponent.  After shrinking the representative,
\begin{equation}\label{analytic:cofinal}
 \cI^a\subseteq\mathcal J\subseteq\cI,
 \qquad
 \cI^{ar}\subseteq\mathcal J^r\subseteq\cI^r
 \quad(r\geq1).
\end{equation}
Such a shrinking may be made on $X$: for a proper map, every open
neighborhood of its compact fiber contains the inverse image of a
sufficiently small neighborhood of the image point.

The proper analytic direct-image theorem makes
$R^1\pi_*\mathcal E$ coherent.  The analytic theorem on formal
functions identifies its completed stalk with
\begin{equation}\label{analytic:formal-functions}
 \widehat{(R^1\pi_*\mathcal E)_x}
 \simeq
 \varprojlim_{r\geq1}H^1(Y,\mathcal E/\mathcal J^r\mathcal E).
\end{equation}
We use \cite[\S6, Haupts\"atze~I and~IIa,
p.~59]{Grauert1960}: the hypotheses are a proper holomorphic map
and a coherent analytic sheaf.  In particular no flatness of
$\pi$ is required.  The quotients in
\eqref{analytic:formal-functions} are supported on the infinitesimal
fibers over $x$, so their cohomology is independent of a further
shrinking of the representative.

For clarity, the right side vanishes by an explicit factorization,
without identifying the individual cohomology groups for the two
filtrations.  The inclusions in \eqref{analytic:cofinal} give
\[
 \mathcal J^{ar}\mathcal E
 \subseteq\cI^{ar}\mathcal E
 \subseteq\mathcal J^r\mathcal E.
\]
Hence the transition map from index $ar$ to index $r$ in
\eqref{analytic:formal-functions} factors through
\[
 H^1(Y,\mathcal E/\cI^{ar}\mathcal E)=0
\]
by Lemma~\ref{analytic:graded-vanishing}.  The inverse system is
therefore pro-zero, and its inverse limit is zero.  Since the stalk
$(R^1\pi_*\mathcal E)_x$ is a finite module over the Noetherian
local ring $\OO_{X,x}$, its zero completion implies that the stalk
itself is zero, by Nakayama's lemma.  Away from $x$ the map $\pi$
is an isomorphism, so the same higher direct image vanishes there.
We obtain $R^1\pi_*\mathcal E=0$.

The sheaf $\pi_*\mathcal E$ is also coherent.  Cartan's
Theorem~B on the Stein space $X$, together with the Leray spectral
sequence for $\pi$, now gives
\[
 H^1(Y,\mathcal E)=0.
\]
The cohomology sequence of
\[
 0\longrightarrow\mathcal E\longrightarrow\cF
 \longrightarrow\cF/\cI^9\cF\longrightarrow0
\]
therefore lifts both $\bar v_i$ to actual holomorphic sections $v_i$
of $\cF$.

It remains to check that arbitrary such lifts retain the required
frame.  In each divisor chart, let $v_i^{(8)}$ be the polynomial
representative of the prescribed jet.  It belongs to $\cI\cF$, and
\[
 v_i-v_i^{(8)}\in\cI^9\cF.
\]
Thus the change in their wedge belongs to
$\cI^{10}\det\cF$.  By Lemma~\ref{analytic:realization}, the
logarithmic determinant of the lifted pair is still
$t^4(1+O(t))$; its leading coefficient is $1$ even at the common
zeros of the leading row $r$.  Its ordinary determinant is
\begin{equation}\label{analytic:ordinary-determinant}
 v_1\wedge v_2=t^5 u(t,z)\,\partial_t\wedge\partial_z,
 \qquad u(0,z)=1.
\end{equation}
The unit condition holds on a neighborhood of each point of $C$.
Compactness gives one neighborhood on which all these local factors
are nonvanishing.  Properness allows a further shrinking of $X$
whose inverse image lies in this neighborhood.  Equation
\eqref{analytic:ordinary-determinant} then proves that $v_1,v_2$
frame $\cT_{Y\setminus C}$.
\end{proof}

\subsection{The singular point and its derivations}

\begin{proposition}\label{prop:analytic-counterexample}
There is a normal complex surface germ $(X,x)$ with an isolated
singular point and free analytic derivation sheaf of rank two:
\[
 \cT_X=\mathcal{H}om_{\OO_X}(\Omega_X^1,\OO_X)
 \simeq\OO_X^{\oplus2}.
\]
It has a resolution $\pi:Y\to X$ whose exceptional set is the
smooth curve $C$ of genus $21$ and self-intersection $-10$.
\end{proposition}

\begin{proof}
Take the contraction constructed in
Lemma~\ref{analytic:actual-lifting}.  Suppose first that $x$ were a
smooth point.  A sufficiently small neighborhood of $x$ would admit
a nowhere-vanishing holomorphic two-form $\omega$.  Its pullback
would be holomorphic on $Y$ and nonvanishing off $C$.  Thus
\[
 \operatorname{div}(\pi^*\omega)=aC\qquad(a\geq0),
 \quad\hbox{and hence}\quad K_Y\cdot C=aC^2\leq0.
\]
Adjunction, however, gives
\[
 K_Y\cdot C=2g(C)-2-C^2=40+10=50,
\]
a contradiction.  The point $x$ is singular, and it is the only
singular point after the chosen shrinking.

Write $X^\circ=X\setminus\{x\}$ and
$j:X^\circ\hookrightarrow X$.  The two fields of
Lemma~\ref{analytic:actual-lifting} descend to a holomorphic frame
$w_1,w_2$ on $X^\circ$.  Normality in dimension two gives the
Hartogs extension property~\cite[Chapter~II, Corollary~7.8]{Demailly2012}
\begin{equation}\label{analytic:hartogs}
 \OO_X\simeq j_*\OO_{X^\circ}.
\end{equation}
This property extends the fields as derivations, as follows.  Embed
a representative of $(X,x)$ in a smooth analytic germ with
coordinate functions $x_1,\ldots,x_e$.  For each $i$ and $\alpha$,
the holomorphic function $w_i(x_\alpha)$ on $X^\circ$ extends
uniquely to a function $a_{i\alpha}$ on $X$.  If $f$ is any local
defining equation of the embedding, then
\[
 \sum_{\alpha=1}^e
 a_{i\alpha}\frac{\partial f}{\partial x_\alpha}=0
\]
on $X^\circ$ and hence on $X$, by density and reducedness.  The
tuple $(a_{i\alpha})_\alpha$ consequently defines a section of
$\mathcal{H}om_{\OO_X}(\Omega_X^1,\OO_X)$ extending $w_i$.  We
continue to denote that extension by $w_i$.

For any local analytic derivation $w$ on $X$, the frame on
$X^\circ$ gives unique holomorphic coefficients $h_1,h_2$ there
such that $w=h_1w_1+h_2w_2$.  Both coefficients extend by
\eqref{analytic:hartogs}.  The extended equality holds on $X$,
since applying its difference to the embedding coordinates gives
holomorphic functions vanishing on the dense open set $X^\circ$.
This proves surjectivity of the map
\[
 \OO_X^{\oplus2}\longrightarrow\cT_X,
 \qquad(h_1,h_2)\longmapsto h_1w_1+h_2w_2.
\]
If a pair lies in its kernel, both coefficients vanish on
$X^\circ$ because $w_1,w_2$ are a frame there, and therefore vanish
on $X$.  The map is also injective, proving the assertion.
\end{proof}

\begin{remark}\label{analytic:invariants}
The construction also determines its canonical discrepancy and a
lower bound for its geometric genus.  The two-form $\omega^\circ$
dual to the frame $w_1,w_2$ generates the canonical sheaf on
$X^\circ$.  Extending reflexively across the normal surface point
gives $\omega_X\simeq\OO_X$; equivalently, we may choose the
canonical divisor $K_X=0$.  Since a normal surface is Cohen--Macaulay,
this reflexive canonical sheaf is its dualizing sheaf, so $X$ is
also Gorenstein; this surface criterion is recalled in
\cite[\S2.4, Definition~2.18 and the following paragraph]{FlennerZaidenberg2001}.
By
\eqref{analytic:ordinary-determinant}, its pullback is
\[
 \pi^*\omega^\circ=t^{-5}u(t,z)^{-1}\,dt\wedge dz.
\]
It is meromorphic on $Y$ with divisor exactly $-5C$.  Thus
\[
 K_Y=\pi^*K_X-5C,
\]
so the discrepancy of $C$ is $-5$.

For the geometric genus, put
$\OO_{nC}=\OO_Y/\cI^n$.  Riemann--Roch and the established
equality $h^0(C,L)=2$ give
\[
 h^1(C,L)=h^0(C,L^3)=12.
\]
In the sequence
\[
 0\longrightarrow L\longrightarrow\OO_{2C}
 \longrightarrow\OO_C\longrightarrow0,
\]
the constant section of $\OO_C$ lifts to $\OO_{2C}$.  Since
$H^2(C,L)=0$, its cohomology sequence gives
\[
 h^1(\OO_{2C})=h^1(C,L)+h^1(C,\OO_C)=12+21=33.
\]
For every $n\geq1$, the sequence with kernel $L^n$ similarly shows
that $H^1(\OO_{(n+1)C})\to H^1(\OO_{nC})$ is surjective.
The theorem on formal functions, with the cofinal filtrations in
\eqref{analytic:cofinal}, identifies their inverse limit with the
completion of $(R^1\pi_*\OO_Y)_x$.  This stalk has finite length,
since the coherent higher direct image is supported at $x$, and
thus equals its completion.  The inverse limit maps onto
$H^1(\OO_{2C})$, proving
\[
 p_g(X,x)=\dim_\CC (R^1\pi_*\OO_Y)_x\geq33.
\]
The exceptional curve has genus $g=21$ and its dual graph has first
Betti number $b=0$.  Consequently $p_g-g-b\geq12$.
\end{remark}
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//05-analytic.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//06-algebraization.tex ====
\section{Algebraization and the affine counterexample}
\label{algebra:section}

We now pass from the analytic germ to an affine variety.  Approximating
the defining equations does not by itself retain their relations.
We approximate the equations and a finite relation matrix together;
their exact zero product, together with sufficiently close approximation,
will recover the same complete local ring.  Tangent freeness then descends
from the continuous derivations of that ring.  This follows the
equation-and-relation determinacy method of Hironaka
\cite[\S3, Theorem~3.3]{Hironaka1969}, used in Artin's
isolated-singularity algebraization
\cite[Theorems~3.8--3.9]{Artin1969}.  We prove the finite-determinacy
statement needed here, including fixed estimates that avoid a
complete-intersection hypothesis.

\subsection{Finite determinacy of equations with relations}

Put
\[
 P=\CC[[x_1,\ldots,x_e]],\qquad
 \mathfrak n=(x_1,\ldots,x_e),\qquad
 I=(f_1,\ldots,f_s),\qquad R=P/I,
\]
where $f=(f_1,\ldots,f_s)\in\mathfrak nP^s$ is a row.  Write
$\mathfrak m_R=\mathfrak nR$, and let an overbar denote reduction modulo $I$.
Choose an $s\times t$ matrix $D$ whose columns generate all the relations
on $f$.  Thus
\begin{equation}
\label{algebra:ideal-presentation}
 P^t\xrightarrow{D}P^s\xrightarrow{f}I\longrightarrow0
\end{equation}
is exact.  Define
\[
 \psi:R^s\longrightarrow R^t,\qquad v\longmapsto v\bar D,
 \qquad K=\ker\psi,
 \qquad
 J=\sum_{j=1}^e R\,\overline{\partial_j f}.
\]
Differentiating $fD=0$ and reducing modulo $I$ shows that $J\subset K$.
Right-exact reduction of \eqref{algebra:ideal-presentation} gives a
presentation of $I/I^2$; its dual therefore identifies
\begin{equation}
\label{algebra:conormal-dual}
 K=\Hom_R(I/I^2,R),\qquad T^1=K/J.
\end{equation}
In particular, no exactness on the left after reduction has been assumed.

\begin{lemma}[Finite determinacy with a relation matrix]
\label{algebra:finite-determinacy}
Suppose that $T^1$ in \eqref{algebra:conormal-dual} has finite length.
There exist integers $N,n_0$ depending only on $f,D$ such that the following
holds.  If a row $g\in P^s$ and an $s\times t$ matrix $B$ satisfy
\[
 gB=0,\qquad
 g-f\in\mathfrak n^{n_0}P^s,\qquad
 B-D\in\mathfrak n^NP^{s\times t},
\]
then there are a continuous $\CC$-algebra automorphism $\Phi$ of $P$ and
an invertible matrix $U\in\operatorname{GL}_s(P)$ such that
\[
 f=\Phi(g)U.
\]
One can take $\Phi(x_j)\equiv x_j\pmod{\mathfrak n^2}$ and
$U\equiv1\pmod{\mathfrak n}$.  Consequently $P/(g)\simeq P/(f)$ as
complete local $\CC$-algebras.
\end{lemma}

\begin{proof}
We improve the equations one order at a time.  The exact relation
$gB=0$ makes the equation error, reduced modulo $I$, nearly belong to
$K$.  Artin--Rees lets us correct it into $K$; the finite length of
$K/J$ then expresses the sufficiently high-order corrected class by
coordinate derivatives.  A change of generators removes the remaining
$I$-valued error.  We arrange these corrections so that the equation
precision increases at every step while the relation precision stays
fixed.  The resulting coordinate and generator changes converge in $P$.

We first choose the fixed bounds. For a map
$u:M\to M'$ of finite modules over a Noetherian local ring with maximal
ideal $\mathfrak a$, the Artin--Rees lemma~\cite[Tag~00IN]{StacksCompletion}, applied to
$\operatorname{im}u\subset M'$ gives a constant $a\ge0$ such that
\[
 \operatorname{im}u\cap\mathfrak a^rM'
 \subset u(\mathfrak a^{r-a}M)\qquad(r\ge a).
\]
Choose nonnegative constants $\alpha,\beta,d$ so that
\begin{align}
 \operatorname{im}\psi\cap\mathfrak m_R^rR^t
 &\subset\psi(\mathfrak m_R^{r-\alpha}R^s),
 \label{algebra:ar-image}\\
 K\cap\mathfrak m_R^rR^s
 &\subset\mathfrak m_R^{r-\beta}K,
 \label{algebra:ar-kernel}\\
 I^{\oplus s}\cap\mathfrak n^rP^s
 &\subset\{fC:C\in\mathfrak n^{r-d}P^{s\times s}\}
 \label{algebra:ar-ideal}
\end{align}
whenever the corresponding exponents are nonnegative.  In
\eqref{algebra:ar-ideal}, Artin--Rees is applied to the map
$P^{s\times s}\to P^s$, $C\mapsto fC$, with image $I^{\oplus s}$; it is the
filtration induced from $P^s$ that is relevant.

Choose $\ell\ge0$ with $\mathfrak m_R^\ell K\subset J$ and set
\begin{equation}
\label{algebra:constants}
 c_0=\beta+\ell,\qquad c=c_0+d,\qquad
 N\ge\alpha+1,\qquad
 n_0\ge\max\{c+N,c+2,2c+1\}.
\end{equation}
We first describe one improvement step.  Suppose $n\ge n_0$ and
\[
 h=g-f\in\mathfrak n^nP^s,
 \qquad B-D\in\mathfrak n^NP^{s\times t},
 \qquad gB=0.
\]
The exact relation, reduced modulo the fixed ideal $I$, gives
\begin{equation}
\label{algebra:high-order-relation}
 \bar h\bar D
 =\bar g\bar D
 =-\bar g(\bar B-\bar D)
 =-\bar h(\bar B-\bar D)
 \in\mathfrak m_R^{n+N}R^t.
\end{equation}
The factor $\bar h$ on the right is what provides order $n+N$.
By \eqref{algebra:ar-image}, there is
\[
 \bar\epsilon\in\mathfrak m_R^{n+N-\alpha}R^s
 \subset\mathfrak m_R^{n+1}R^s,
 \qquad \psi(\bar\epsilon)=\psi(\bar h).
\]
Hence
\[
 \kappa=\bar h-\bar\epsilon
 \in K\cap\mathfrak m_R^nR^s
 \subset\mathfrak m_R^{n-\beta}K
 \subset\mathfrak m_R^{n-c_0}J.
\]
Here the last inclusion uses $\mathfrak m_R^\ell K\subset J$,
with the remaining factor of order $n-\beta-\ell=n-c_0$.
Lift coefficients expressing $\kappa$ in the displayed generators of
$J$ to elements $a_j\in\mathfrak n^{n-c_0}$, and lift
$\bar\epsilon$ to $\epsilon\in\mathfrak n^{n+1}P^s$.  Then
\[
 w=h-\sum_{j=1}^e a_j\partial_jf-\epsilon
 \in I^{\oplus s}\cap\mathfrak n^{n-c_0}P^s.
\]
The order of this ideal-valued remainder need only be $n-c_0$, since
the lifted derivative combination may have lower order than $h$:
a derivative of a defining equation can be a unit.
Using \eqref{algebra:ar-ideal}, write $w=fC$ with
$C\in\mathfrak n^{n-c}P^{s\times s}$.  We have proved
\begin{equation}
\label{algebra:linear-correction}
 h=\sum_{j=1}^e a_j\partial_jf+fC+\epsilon,
 \qquad
 a_j,C\in\mathfrak n^{n-c},
 \qquad \epsilon\in\mathfrak n^{n+1}P^s.
\end{equation}
Thus the loss $d$ for lifting the ideal-valued remainder has been included
in the fixed constant $c$.

Put $q=n-c$, and define
\begin{equation}
\label{algebra:one-step}
 \phi(x_j)=x_j-a_j,\qquad M=(1+C)^{-1},\qquad
 g_{\mathrm{new}}=\phi(g)M,\qquad
 B_{\mathrm{new}}=(1+C)\phi(B).
\end{equation}
Our bounds ensure
\[
 q\ge\max\{N,2\},\qquad 2q\ge n+1,
 \qquad n+q-1\ge n+1.
\]
In particular, $\phi$ is a formal automorphism with identity linear part,
and $M$ is invertible.  Formal Taylor expansion gives
\[
 \phi(f)-f+\sum_j a_j\partial_jf\in\mathfrak n^{2q}P^s,
 \qquad
 \phi(h)-h\in\mathfrak n^{n+q-1}P^s.
\]
Combining these estimates with \eqref{algebra:linear-correction} gives
\[
 \phi(g)\equiv f(1+C)\pmod{\mathfrak n^{n+1}P^s},
 \qquad
 g_{\mathrm{new}}-f\in\mathfrak n^{n+1}P^s.
\]
The relations in \eqref{algebra:one-step} transform exactly:
\[
 g_{\mathrm{new}}B_{\mathrm{new}}
 =\phi(g)(1+C)^{-1}(1+C)\phi(B)=0.
\]
Moreover, substitution by $\phi$ changes any power series by an element
of $\mathfrak n^q$, and $C$ has order at least $q$.  Therefore
\[
 B_{\mathrm{new}}-D
 =C\phi(B)+(\phi(B)-B)+(B-D)
 \in\mathfrak n^NP^{s\times t}.
\]
The fixed relation precision $N$ is preserved, while the equation
precision increases from $n$ to $n+1$.

Starting with $(g_{n_0},B_{n_0})=(g,B)$, repeat this step for
$n=n_0,n_0+1,\ldots$.  Write $\phi_n,M_n$ for its corrections.
For completeness, the cumulative coordinate and generator changes obey
\[
 \Phi_{n_0}=\id,\quad U_{n_0}=1,\qquad
 \Phi_{n+1}=\phi_n\circ\Phi_n,\qquad
 U_{n+1}=\phi_n(U_n)M_n,
\]
so that $g_n=\Phi_n(g)U_n$ at every stage.  Since
\[
 (\phi_n-\id)(P)\subset\mathfrak n^{n-c},
 \qquad M_n-1\in\mathfrak n^{n-c}P^{s\times s},
\]
the consecutive images $\Phi_n(x_j)$ and the consecutive matrices
$U_n$ differ in orders tending to infinity.  They converge in the
complete ring $P$ to $\Phi(x_j)$ and $U$.  The resulting substitution
$\Phi$ has identity linear part and hence is an automorphism, while
$U\equiv1\pmod{\mathfrak n}$ is invertible.  Taking limits in
$g_n=\Phi_n(g)U_n$ yields $f=\Phi(g)U$, as required.
\end{proof}

The original matrix $D$ must generate all the relations, in order that
\eqref{algebra:conormal-dual} hold.  No such assertion is needed for the
approximating matrix $B$: its exact zero product and its closeness to
$D$ are sufficient.  Thus Lemma~\ref{algebra:finite-determinacy} concerns
solutions of the finite system $gB=0$, not arbitrary truncations of a
list of defining equations.

\subsection{The algebraic neighborhood}

\begin{proposition}[Algebraization of an isolated analytic germ]
\label{prop:algebraization}
Let $(X,x)$ be a normal complex analytic germ of dimension $d\ge1$,
smooth away from $x$, and suppose that its holomorphic tangent module
is free of rank $d$.  There exist a finitely generated normal integral
$\CC$-algebra $A$ and a maximal ideal $\mathfrak p\subset A$ such that
\[
 \dim A=d,\qquad A/\mathfrak p=\CC,\qquad
 \widehat{A_{\mathfrak p}}\simeq\widehat{\OO_{X,x}},\qquad
 \Der_{\CC}(A)\simeq A^d.
\]
The affine variety $\Spec A$ can be chosen smooth away from
$\mathfrak p$.  If $x$ is singular, $A_{\mathfrak p}$ is nonregular.
\end{proposition}

\begin{proof}
\emph{The completed analytic presentation.}
Embed the germ in a smooth analytic germ and write
\[
 S=\OO_{X,x}=\CC\{x_1,\ldots,x_e\}/I_{\mathrm{an}}.
\]
Choose a finite row $f$ generating $I_{\mathrm{an}}$ and a finite matrix
$D$ generating all its relations over the convergent power-series ring.
Completion is exact on finite modules, so they give the presentation
\eqref{algebra:ideal-presentation} over $P=\CC[[x_1,\ldots,x_e]]$,
with $R=\widehat S=P/(f)$.  Complex analytic local rings are excellent
\cite[Proposition~2.1(1)]{Cutkosky2017};
thus the completion of the normal local ring $S$ is a normal local
domain.  Also
\begin{equation}
\label{algebra:completion-invariants}
 \dim R=\dim S=d,
 \qquad R\text{ is regular if and only if }S\text{ is regular}.
\end{equation}
We use here the usual exactness and permanence properties of completion
\cite{StacksCompletion}.

The convergent version of \eqref{algebra:conormal-dual} is the conormal
dual of the analytic embedding.  On the smooth locus the tangent-to-normal
map is surjective: a smooth analytic subspace of a complex manifold is
locally a regular embedding.  Its cokernel is consequently a coherent
module supported at the isolated point $x$, and has finite length.
Flat completion identifies this cokernel with the module $T^1=K/J$
above.  Lemma~\ref{algebra:finite-determinacy} therefore applies to $f,D$.

We also record precisely which differentials are being completed.  The
module of holomorphic differentials of $S$ has the finite Jacobian
presentation with generators $dx_j$ and relations $df_i$.  Its dual is
the holomorphic tangent module.  Flat completion of this kernel yields
\begin{equation}
\label{algebra:continuous-tangent}
 \Der^{\mathrm{cont}}_{\CC}(R,R)
 =\ker\left(R^e\longrightarrow R^s,
       (v_j)\longmapsto\sum_{j=1}^e v_j\overline{\partial_jf}\right)
 \simeq R^d.
\end{equation}
Here ``continuous'' refers to the maximal-ideal topology.  A continuous
derivation of a power-series quotient is determined by its values on
the $x_j$, and the displayed relations are precisely the conditions for
it to descend from $P$.  All completed differential modules used below
are these finite continuous differential modules.

\emph{Approximation of equations and relations.}
Let
\[
 H=\bigl(\CC[x_1,\ldots,x_e]_{(x_1,\ldots,x_e)}\bigr)^h
\]
be the henselization, whose completion is $P$.  This is precisely a
henselization of a finite-type algebra over the field $\CC$ at a prime,
as covered by Artin's Theorem~1.10.  Apply that approximation theorem
over $H$ to the polynomial system in the $s+st$ entries of a row $G$
and a matrix $B$,
\begin{equation}
\label{algebra:artin-system}
 \sum_{i=1}^s G_i B_{i\nu}=0\qquad(1\le\nu\le t),
\end{equation}
with formal solution $(G,B)=(f,D)$.  It gives a row $g$ and a matrix $B$
with entries in $H$, satisfying $gB=0$ exactly and agreeing with $f,D$
to any prescribed finite order
\cite[Theorem~1.10]{Artin1969}; see also~\cite{StacksApproximation}.  Choose that
order at least $\max\{n_0,N\}$, with $n_0,N$ from
Lemma~\ref{algebra:finite-determinacy}.  The lemma gives
\begin{equation}
\label{algebra:formal-equivalence}
 P/(g)\simeq R.
\end{equation}

The henselization is the filtered colimit of pointed \'etale
neighborhoods of the origin, with the same residue field
\cite{StacksHenselization}.  Thus the finitely many entries of $g,B$
are represented on a single such neighborhood.  After passing to a
later neighborhood, the finitely many identities
\eqref{algebra:artin-system} hold there as well.  If this stage is
expressed over the local polynomial ring, finite presentation and
clearing finitely many denominators spread it to a neighborhood of
the origin in affine space; shrinking preserves the marked point
and makes the morphism \'etale.  After an affine shrinking the
identities hold in its coordinate ring.  We obtain an affine
\'etale neighborhood $\Spec E\to\mathbb A^e_{\CC}$ with a marked point
$\mathfrak q$ over the origin, residue field $\CC$, and a row $g$ of
elements of $E$ vanishing at $\mathfrak q$.  Its completed ambient
local ring is $P$.  Put
\[
 A'=E/(g),\qquad \mathfrak p'=\mathfrak q/(g),\qquad
 T=A'_{\mathfrak p'}.
\]
Then $A'$ is of finite type over $\CC$, and exactness of completion
together with \eqref{algebra:formal-equivalence} gives
\begin{equation}
\label{algebra:completed-local-ring}
 \widehat T=P/(g)\simeq R.
\end{equation}
In particular, this construction preserves the complete local ring,
including its dimension and its normality.

\emph{Descent of tangent freeness and local ring properties.}
Because the ambient neighborhood is \'etale over $\mathbb A^e_{\CC}$,
its differentials have basis $dx_1,\ldots,dx_e$ near the marked point.
Completing the finite presentation of $\Omega^1_{T/\CC}$ therefore
gives the continuous Jacobian presentation of $P/(g)$.  For a finitely
presented module, flat base change commutes with its dual: this follows
by expressing the dual as the kernel of a map between finite free
modules.  The continuous chain rule under
\eqref{algebra:completed-local-ring} and
\eqref{algebra:continuous-tangent} consequently give
\begin{equation}
\label{algebra:tangent-descent}
 \Der_{\CC}(T)\otimes_T\widehat T
 \simeq\Der^{\mathrm{cont}}_{\CC}(\widehat T,\widehat T)
 \simeq\widehat T^{d}.
\end{equation}
This is an equality obtained from finite presentations; it does not
identify ordinary algebraic differentials of a power-series ring with
its continuous differentials.

Let $M=\Der_{\CC}(T)$.  By \eqref{algebra:tangent-descent}, its residue
vector space has dimension $d$.  Lift a basis to $d$ elements of $M$.
Nakayama's lemma gives a surjection $T^d\to M$, which after completion
is a surjection between free modules of rank $d$ over the local ring
$\widehat T$, and hence an isomorphism.  Faithful flatness of completion
annihilates its kernel, proving $M\simeq T^d$.

The ring $T$ injects into its domain completion, so it is a domain.
Normality also descends.  Indeed, if $a/b\in\operatorname{Frac}(T)$
is integral over $T$, it is integral over the normal domain
$\widehat T$ and hence belongs to $\widehat T$.  Thus
$a\in b\widehat T\cap T=bT$, where the last equality follows from
faithful flatness, and $a/b\in T$.  Finally, completion preserves both
Krull dimension and the dimension of the maximal ideal modulo its
square.  Equations \eqref{algebra:completion-invariants} and
\eqref{algebra:completed-local-ring} show that $T$ has dimension $d$
and is nonregular whenever $S$ is nonregular.

The isolated-singularity property can also be retained.  Let
$\mathfrak j_{\mathrm{an}}$ be the $d$th Fitting ideal of the
holomorphic differential module of $S$.  The analytic Jacobian
criterion identifies its zero set with the nonsmooth locus, so some
power of the maximal ideal of $S$ is contained in
$\mathfrak j_{\mathrm{an}}$; in the smooth case this ideal is the unit
ideal.  Fitting ideals commute with base change.  The completed
Jacobian presentations and \eqref{algebra:completed-local-ring} thus
imply, by contraction from the faithfully flat completion, that
\begin{equation}
\label{algebra:isolated-descent}
 \mathfrak m_T^b\subset
 \operatorname{Fitt}_d(\Omega^1_{T/\CC})
 \quad\text{for some }b\ge0.
\end{equation}
On the reduced pure-$d$-dimensional neighborhood constructed below,
the algebraic Jacobian criterion interprets this containment as
nonsmoothness supported at the marked point.

\emph{An integral normal affine neighborhood with a free tangent module.}
The finite module $\Der_{\CC}(A')$ localizes at $\mathfrak p'$ to
$\Der_{\CC}(T)$, because $\Omega^1_{A'/\CC}$ is finitely presented.
Represent the basis just obtained after inverting an element outside
$\mathfrak p'$.  It defines a map from a free module of rank $d$ to
the derivation module whose kernel and cokernel vanish at
$\mathfrak p'$.  Their supports are closed, so a further principal
shrinking makes this map an isomorphism.

Since $T$ is a domain, exactly one irreducible component of $\Spec A'$
passes through $\mathfrak p'$, and the support of the nilradical avoids
$\mathfrak p'$.  Remove the other components and that support.  The
remaining neighborhood is reduced and irreducible.  Its normal locus
is open, since schemes of finite type over $\CC$ are excellent, and
contains $\mathfrak p'$ by the normality of $T$; restrict to this
normal locus as well.  This irreducible neighborhood has dimension
$d$, since its marked point is closed and its local ring has dimension
$d$.  Its nonsmooth locus is closed and, by
\eqref{algebra:isolated-descent}, has no positive-dimensional component
through $\mathfrak p'$.  Remove any components of that locus that do
not contain $\mathfrak p'$.

All the restrictions just made are open neighborhoods of
$\mathfrak p'$.  Choose a principal affine neighborhood contained in
their intersection, and write it as $\Spec A$, with marked maximal
ideal $\mathfrak p$.  The algebra $A$ is finitely generated, normal
and integral, its dimension is $d$, and
\[
 A_{\mathfrak p}=T,\qquad
 \Der_{\CC}(A)\simeq A^d.
\]
The marked residue field and completed local ring are unchanged, and
the variety is smooth away from that point.  This proves all the
assertions.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Proposition~\ref{prop:analytic-counterexample} supplies a singular
normal complex analytic surface germ, smooth away from its marked
point, whose holomorphic tangent module is free of rank two.
Apply Proposition~\ref{prop:algebraization} and fix any of the
approximations and affine neighborhoods furnished by its proof.  Its
coordinate ring $A$ is a finitely generated normal integral
$\CC$-algebra of dimension two, with
\[
 \Der_{\CC}(A)\simeq A^2,
 \qquad A_{\mathfrak p}\text{ nonregular}.
\]
Thus the derivation module is free, and in particular projective,
whereas $A$ is not regular.  This is the required counterexample over
$\CC$.
\end{proof}

\subsection{The smooth locus and derivation blowups}

Let $X=\Spec A$ be the affine surface just constructed, and let
$U=X_{\mathrm{reg}}$ be its smooth locus.  Recall that the logarithmic
Kodaira dimension of a smooth quasiprojective surface is
\[
 \overline\kappa(U)=\kappa(K_{\overline U}+D),
\]
where $\overline U$ is a smooth projective compactification and its
reduced boundary $D=\overline U\setminus U$ is a simple normal-crossing
divisor.  This value is independent of the compactification.

\begin{corollary}\label{algebra:log-kodaira}
The smooth locus $U$ of the affine surface $X=\Spec A$ constructed in
Theorem~\ref{thm:main} has $\overline\kappa(U)=2$.
\end{corollary}

\begin{proof}
The free tangent sheaf on $X$ restricts to a trivial tangent bundle on
$U$.  If $\overline\kappa(U)\le1$, the theorem of Biswas, Gurjar and
Kolte~\cite[Theorem~0.1(1)]{BiswasGurjarKolte2014} would make the normal
affine surface $X$ smooth, contrary to Theorem~\ref{thm:main}.
The only remaining value of logarithmic Kodaira dimension for a surface
is two.
\end{proof}

For a finite $A$-module $M$ of generic rank $r$, viewed as a coherent
sheaf on $X$, its blowup is universal among proper birational morphisms
from integral varieties to $X$ for which the pullback of $M$ modulo
torsion is locally free of rank $r$.  Every such morphism factors uniquely
through the blowup.  We use Villamayor's description, as recalled in
\cite[\S1.6, Theorem~1.17]{BarajasChavezRomano2026}.

\begin{corollary}\label{algebra:derivation-blowups}
Starting with the singular surface $X=\Spec A$ of
Theorem~\ref{thm:main}, every iteration of blowing up at the derivation
module is isomorphic to $X$.  The same holds if each blowup is followed
by normalization.  Neither procedure resolves its singular point.
\end{corollary}

\begin{proof}
Since $\Der_{\CC}(A)\simeq A^2$, the identity of $X$ already has the
universal property just stated.  Its blowup at the derivation module
is therefore $X$ itself, and the operation remains unchanged at every
iteration.  Normalization also leaves $X$ unchanged because $A$ is
normal.  The relation with the Lipman--Zariski conjecture was noted in
\cite[Introduction, after Question~1]{BarajasChavezRomano2026}; the
surface above gives a negative answer to that question in characteristic
zero.
\end{proof}
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//06-algebraization.tex ====

==== BEGIN A-singular-normal-affine-surface-with-free-tangent-sheaf//bibliography.tex ====
\begin{thebibliography}{99}

\bibitem{Artin1969}
M.~Artin,
\emph{Algebraic approximation of structures over complete local rings},
Publ. Math. IH\'ES \textbf{36} (1969), 23--58.
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\end{thebibliography}
==== END A-singular-normal-affine-surface-with-free-tangent-sheaf//bibliography.tex ====

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