The Lipman-Zariski conjecture predicts that a complex algebraic
variety X of dimension n is smooth whenever its tangent sheaf is free
of rank n. Equivalently, a finitely generated integral complex algebra
A whose module Der(A) of complex derivations is a free A-module of
rank equal to dim(A) is regular. The positive characteristic analogue
fails: over a perfect field of characteristic p the affine surface
xy - z^p = 0 has tangent sheaf freely generated by x d_x - y d_y and
d_z, while the local ring at the origin is not regular (Graf, J.
Algebra 601, 2022, 115-128).

./graf-2201.06646-positive-characteristic.pdf

Over the complex numbers the answer is also negative. There exist a
finitely generated normal integral complex algebra A of Krull
dimension two and a maximal ideal m of A such that Der(A) is
isomorphic to A^2 and A_m is not regular. The affine surface Spec(A)
can be chosen smooth away from m. Its unique singular point has
geometric genus at least 33 and canonical discrepancy -5, and the
resolution pi: Y -> X has one exceptional curve C that is smooth of
genus 21 and self-intersection -10.

Prior work narrows the possibilities. Lipman showed that local
freeness of the tangent sheaf forces normality (Lipman, Amer. J.
Math. 87, 1965, 874-898). For positively graded complex algebras with
isolated singularity and degree-zero part the base field, Hochster
proved that freeness of the derivation module at the irrelevant
maximal ideal forces the algebra to be a polynomial algebra
(Hochster, J. Algebra 47, 1977, 411-424). Hypersurfaces were covered
by Scheja and Storch (Math. Ann. 197, 1972, 137-170). Algebraic
locally complete intersections in characteristic zero were covered by
Kallstrom (J. Algebra 337, 2011, 169-180; corrected version
arXiv:1003.4241v6, 2018).

./kallstrom-1003.4241-complete-intersections.pdf

Extension of holomorphic differential forms gives the conjecture for
isolated singularities in dimension at least three (van Straten and
Steenbrink, Abh. Math. Sem. Univ. Hamburg 55, 1985, 97-110). The log
canonical case was obtained independently by Greb, Kebekus, Kovacs
and Peternell (Publ. Math. IHES 114, 2011, 87-169), by Druel (Bull.
London Math. Soc. 46, 2014, 827-835), and by Graf and Kovacs (Doc.
Math. 19, 2014, 815-830).

./gkkp-1003.5340-differential-forms-log-canonical.pdf
./druel-1304.6000-log-canonical-spaces.pdf
./graf-kovacs-1301.7315-optimal-extension-1forms.pdf

For normal complex surfaces, the geometric genus together with the
configuration of the exceptional divisor gives sharp criteria. Graf
established smoothness when the geometric genus is at most one, and
when it equals two provided the exceptional divisor is not a tree of
rational curves (Int. Math. Res. Not. 2021, 426-441). Bergner and
Graf extended the admissible range to p_g - g - b <= 2, where g is
the total genus of the exceptional curves and b is the first Betti
number of the dual graph (Forum Math. Sigma 8, 2020, paper e21).
Biswas, Gurjar and Kolte proved smoothness for normal affine
surfaces whose tangent bundle on the smooth locus is trivial and
whose smooth locus has logarithmic Kodaira dimension at most one (J.
London Math. Soc. 90, 2014, 270-286).

./graf-1801.05753-low-genus.pdf
./bergner-graf-2020-lipman-zariski-genus-one-higher.pdf
./bgk-1403.5613-zariski-lipman-normal-surfaces.pdf

The surface to be constructed lies outside every one of these
positive results. Its canonical discrepancy is -5, so it is not log
canonical. It has one exceptional curve of genus 21, so the
Bergner-Graf invariant p_g - g - b is at least 12. Its smooth locus
has logarithmic Kodaira dimension two. The surface also answers
negatively, in characteristic zero, the question of Barajas,
Chavez-Martinez and Romano-Velazquez (arXiv:2510.07607v2) whether
iterated blowup at the module of derivations resolves a singularity:
on a variety with free tangent sheaf every such blowup, with or
without subsequent normalization, is already the identity.

./barajas-chavez-romano-2510.07607-blowup-derivations.pdf

An additional intrinsic constraint comes from the bracket of any
frame. Jorder proved that a normal complex space whose tangent sheaf
locally admits a basis of pairwise commuting vector fields is smooth
(Math. Z. 278, 2014, 893-899). Hence no frame for the tangent sheaf
of the surface below can be made commuting near its singular point.

./jorder-1311.5141-weak-lipman-zariski.pdf

Conventions. Line bundles on curves are written additively; aL + bH
denotes the tensor product L^a tensor H^b over O_C. Two-dimensional
row vectors X = (X_1, X_2) and Y = (Y_1, Y_2) have wedge product
X /\ Y = X_1 Y_2 - X_2 Y_1. A quadratic form Q on a two-dimensional
space has bilinear polar form x . y = (Q(x+y) - Q(x) - Q(y)) / 2, and
x^2 abbreviates Q(x). A residue on an annular overlap means (1/(2 pi
i)) times a positively oriented contour integral. Adjunction and
residue formulas put the normal differential first: locally
Res_{y=0}((dy/y) /\ dz) = dz. All varieties and germs are complex
analytic or complex algebraic as dictated by context. Background
references are: Hartshorne, Algebraic Geometry, GTM 52 (Springer
1977) for curve cohomology, adjunction and Riemann-Roch; Serre (Ann.
Inst. Fourier 6, 1956, 1-42) for the GAGA comparison theorem;
Grauert and Remmert, Coherent Analytic Sheaves, Grundlehren 265
(Springer 1984), and Demailly, Complex Analytic and Differential
Geometry (2012), for coherent analytic sheaves and normal complex
spaces.

./serre-1956-gaga.pdf
./demailly-2012-complex-analytic-differential-geometry.pdf

The construction begins on the quadric surface Sigma = PP^1 x PP^1
with ruling line bundles F = O(1,0) and S = O(0,1). Let Gamma be a
smooth elliptic curve. Choose a degree-two line bundle F on Gamma
and a line bundle D_0 of exact order four; set S = F + D_0. Both F
and S on Gamma are generated by their two-dimensional spaces of
sections. After choosing an origin, the two fiber involutions take
the form p -> a - p and p -> a + d_0 - p, where d_0 is a nonzero
representative of D_0. Two distinct points cannot be interchanged by
both involutions, and no point is fixed by both. The product map is
therefore an embedding, and its image has bidegree (2,2). The same
symbols F and S denote the ruling line bundles O(1,0) and O(0,1) on
Sigma.

On Gamma put

  Delta = 2 D_0,
  Phi = F + Delta,
  M = 3 Phi.

Delta is nontrivial of order two, and

  Gamma|_Gamma = 2F + 2S = 4F + Delta,
  M = 3F + Delta,
  2M = 6F.

Consequently

  4M = (2 Gamma + 4F)|_Gamma,
  2M - (Gamma + F)|_Gamma = Phi,
  (Gamma + 2F)|_Gamma - 2M = Delta.

Fix isomorphisms in these identities compatibly. In particular, Phi =
2M - (Gamma + F)|_Gamma induces the identification 2 Phi = 2F|_Gamma
used below.

Since K_Sigma + Gamma = 0, there is a nowhere-vanishing logarithmic
two-form Omega on Sigma whose pole divisor is Gamma. Its residue
omega = Res_Gamma(Omega) is a nowhere-vanishing differential on
Gamma. The residue convention places the normal differential first:
locally Res_{y=0}((dy/y) /\ dz) = dz. Choose a basis u = (u_1, u_2)
of H^0(Sigma, F), and set 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. The analogous eta_C is
defined on C below.

There is a unique element of Sym^2 H^0(Gamma, Phi) whose image under
multiplication is

  -eta_Gamma / omega in H^0(Gamma, 2 Phi) = H^0(Gamma, 2 F).

It defines a nondegenerate quadratic form Q on H^0(Gamma, Phi)^*. For
existence and nondegeneracy, let iota_F and iota_Phi be the two
degree-two involutions. One has iota_Phi = t_Delta o iota_F, where
t_Delta is translation by the nontrivial point of order two
represented by 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 that quadratic produces
-eta_Gamma / omega. Uniqueness follows because pullback from the
surjective Phi-map is injective on sections of O_{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 nonzero scalar. The
projective direction is [ev_q]. By the quadratic identity, [ev_q] is
isotropic precisely when q is a ramification point of F|_Gamma. On
every two-dimensional quadratic space the polar form is x . y =
(Q(x+y) - Q(x) - Q(y))/2, and x^2 abbreviates Q(x).

The next lemma produces six vectors in H^0(Gamma, Phi)^* with
prescribed sum and normalization. If w = sum_j w_j and w_j^2 + w_j .
w = 1 at each marked point, then

  sum_{j=1}^6 (w_j^2 - 1) = -w^2.

Since deg L = 10 below, the choice w^2 = 90 gives the total -9 deg L
required by the order-two obstruction discussed later.

Lemma. In a two-dimensional nondegenerate complex quadratic space,
three distinct nonisotropic projective directions can be chosen,
avoiding any prescribed finite set, and nonzero vectors w_1, ..., w_6
using each direction twice, with the two vectors in each pair equal,
such that

  w = sum_{j=1}^6 w_j,   w^2 = 90,   w_j^2 + w_j . w = 1   (1 <= j <= 6).

Proof. Choose coordinates in which Q(x,y) = xy, so the polar form is
(x,y) . (x',y') = (xy' + yx')/2. Set

  h = sqrt(90) > 0,
  R = sqrt(1 + h^2/4) > 0,
  b = 2h/R,
  d(z) = (R z - h/2, R/z - h/2).

Direct expansion gives d(z)^2 + d(z) . (h,h) = R^2 - h^2/4 = 1. For
sufficiently small general complex lambda != 0, let z_1, z_2, z_3
be the roots of

  X^3 - b X^2 + b lambda X - lambda = 0.

Vieta gives sum z_i = b, sum 1/z_i = b, and 2 sum_i d(z_i) = (h,h).
Taking each d(z_i) twice produces the required vectors.

The three directions are distinct. As lambda -> 0, two roots go to
zero and the third goes to b. The roots are distinct for general
lambda: the cubic discriminant has nonzero leading term
-4 b^3 lambda. Near zero, the projective ratio of the entries of d(z)
is z(R z - h/2)/(R - h z/2), whose derivative at zero is
-h/(2R) != 0. The two small roots give distinct nonisotropic
directions. At z = b, the entries are 3h/2 and (R^2 - h^2)/(2h),
both nonzero, so the third direction differs from their common
limit. Apply simultaneously to all six vectors the isometry
(x,y) -> (nu x, nu^{-1} y). On nonisotropic directions the ratio is
multiplied by nu^2, so only finitely many values of nu != 0 are
excluded by the prescribed finite set; all required equations are
preserved. []

Apply the lemma to Q, excluding the branch values of the Phi-map. Each
of the three directions is [ev_q] at two distinct points of Gamma.
Label the six points q_1, ..., q_6 so [w_j] = [ev_{q_j}], and put

  Z = q_1 + ... + q_6 in |3 Phi| = |M|.

Choose a section beta of M with divisor Z. Nonisotropy implies
F|_Gamma is unramified at every q_j. Their F-values are not all
equal, since a degree-two fiber cannot contain six distinct points.

The curve C is constructed with fourth-order contact with Gamma at
every q_j. Let gamma denote the section of O_Sigma(Gamma) defining
Gamma. Choose a section

  kappa in H^0(Sigma, 2 Gamma + 4 F),  kappa|_Gamma = beta^4,

using the fixed identification above. Such sections exist because
restriction to Gamma has kernel O_Sigma(Gamma + 4 F) = O_Sigma(6,2),
with H^1 = 0. The affine space of choices is
kappa_0 + gamma H^0(Sigma, Gamma + 4F). A general choice defines a
smooth curve C: the linear system is base-point-free off Gamma, so
Bertini gives smoothness there; on Gamma minus Z every choice is
nonzero; and at each q_j, global generation of O_Sigma(6,2) makes the
normal derivative vary nontrivially. Avoiding six proper affine
hyperplanes makes those derivatives nonzero. Independence of the six
evaluation functionals is unnecessary.

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 so that k(0,z) = b(z)^4, b'(0) != 0, and
k_y(0,0) != 0. The implicit function theorem writes C locally as
y = phi(z), where

  phi(z) = - b'(0)^4 / k_y(0,0) * z^4 + O(z^5).

Thus the contact order is exactly four, div_C(gamma|_C) = 4 Z, and
the six marked points are read both on Gamma and on C.

The bidegree of C is (8,4). The defining-divisor sequence, with
H^1(Sigma, O_Sigma(-8,-4)) = 0, gives H^0(C, O_C) = C, so C is
connected and irreducible. Put

  H = F|_C,  L = H + Z,  theta = 2L,

and let xi be the canonical section of O_C(Z). The divisor equality
specifies the isomorphism 4 O_C(Z) -> O_C(Gamma) by xi^4 -> gamma|_C.
Together with adjunction this gives

  4L = 4H + 4Z = (Gamma + 4F)|_C = K_C.

The last identification is normalized by the requirement that a local
section s of Gamma + 4F on Sigma maps to

  s|_C -> Res_C( s gamma Omega / kappa ),

with the normal differential first. This fixes the scalar in the
canonical-root isomorphism for subsequent residue computations.

The next lemma computes the section spaces that transfer from Sigma to
C and to Gamma. It uses the comparison theorem of Serre (Ann. Inst.
Fourier 6, 1956, 1-42): algebraic and analytic section spaces of
coherent sheaves on projective complex curves agree.

./serre-1956-gaga.pdf

Lemma (adjoint jets). For 0 <= j <= 3, put A_j = Gamma + (4-j)F =
(6-j, 2) on Sigma and N_j = A_j|_Gamma = (8-j)F + Delta on Gamma.
Restriction identifies H^0(Sigma, A_j) with H^0(C, K_C - jH) and
maps it onto H^0(Gamma, N_j). For 1 <= ell <= 4, evaluation on the
length-ell subschemes at q_1, ..., q_6 has the same rank on C as on
Gamma. The ranks needed below are:

  j = 0, ell = 1: h^0(Gamma, N_0) = 16; N_0 - Z = 5F;
                   h^0(Gamma, 5F) = 10; rank = 6.
  j = 1, ell = 2: h^0(Gamma, N_1) = 14; N_1 - 2Z = Phi;
                   h^0(Gamma, Phi) = 2; rank = 12.
  j = 2, ell = 2: h^0(Gamma, N_2) = 12; N_2 - 2Z = Delta;
                   h^0(Gamma, Delta) = 0; rank = 12.
  j = 3, ell = 2: h^0(Gamma, N_3) = 10; N_3 - 2Z = -F + Delta;
                   h^0(Gamma, -F + Delta) = 0; rank = 10.

The identities use 2Z = 6F, valid because Delta = 2D_0 has order 2.

Proof. The restriction kernels on Sigma are A_j - C = (-2-j, -2)
and A_j - Gamma = (4-j, 0). Product cohomology on PP^1 x PP^1 gives
H^0 = H^1 = 0 for the first kernel and H^1 = 0 for the second, so
restriction H^0(Sigma, A_j) -> H^0(C, K_C - jH) is an isomorphism and
H^0(Sigma, A_j) -> H^0(Gamma, N_j) is onto. By the contact expansion,
the ideal of the length-ell subscheme at q_j in Sigma equals
(y - phi(z), z^ell) = (y, z^ell) for ell <= 4, and the same ideal
defines the length-ell subscheme on C. So the two subschemes agree,
and the ranks of evaluation agree.

On the elliptic curve deg F = 2 and Z = 3F + Delta. Positive-degree
bundles have as many sections as their degree; the nonzero 2-torsion
Delta has none; negative-degree bundles have none. Subtracting the
residual section dimension from h^0(Gamma, N_j) gives the stated
evaluation ranks. []

Proposition (curve data). There exist a smooth connected projective
curve C, a reduced divisor Z = q_1 + ... + q_6 on C, line bundles H,
L, theta on C, and a basis u = (u_1, u_2) of H^0(C, H) with:

(1) g(C) = 21, deg H = 4, deg L = 10, L = H + Z, theta = 2L, and
K_C = 4L under the fixed normalized isomorphism.

(2) The pencil u defines a finite degree-four morphism f: C -> PP^1
unramified at every q_j, with f(q_j) not all equal. The zero divisor
of u_1 can be chosen to be a reduced fiber disjoint from Z.

(3) Section dimensions and multiplication:
  h^0(C, jH) = j+1  (0 <= j <= 3),
  h^0(C, L) = h^0(C, H+2Z) = 2,
  h^0(C, theta) = 3,
  h^0(C, O_C(Z)) = 1,
  h^0(C, theta + H) = 6,
  H^0(C, L) = xi H^0(C, H),
  H^0(C, H+2Z) = xi^2 H^0(C, H),
  H^0(C, theta) = xi^2 H^0(C, 2H).
In particular r = xi u is a basis of H^0(C, L) with common zero
divisor exactly Z.

(4) For every k >= 2, H^0(C, (k+1)L) -> H^0(Z, (k+1)L|_Z) is
surjective.

(5) The auxiliary data on Gamma (bundle identities, Omega, Q, and
the six vectors w_j) satisfy w_j^2 + w_j . w = 1 and w^2 = 90 =
9 deg L; and Z belongs to |M|.

Proof. Throughout, C has bidegree (8,4) and K_C = (6,2)|_C. By
intersection theory on the quadric,
  deg H = (1,0).(8,4) = 4,
  deg K_C = (6,2).(8,4) = 40,
  g(C) = 21.
Since deg Z = 6, deg L = 10. The morphism f given by u is the first
projection restricted to the irreducible C, so it is finite of degree
four. Tangency at q_j identifies the differential of f there with
that of F|_Gamma, which is nonzero. The marked values f(q_j) equal
the F|_Gamma-values at q_j, which are not all equal, so a regular
fiber away from them exists. Replacing u by a determinant-one change
of basis makes that fiber the zero divisor of u_1 and preserves
eta_C and eta_Gamma.

For 0 <= j <= 3, the defining-divisor sequence for C tensored with
O_Sigma(j, 0) has kernel O_Sigma(j-8, -4), whose cohomology vanishes
by product formula, so h^0(C, jH) = h^0(Sigma, O(j, 0)) = j + 1.

For a line bundle B and effective divisor D on C, Riemann-Roch
followed by Serre duality gives the dimension formula:
  h^0(C, B+D) - h^0(C, B)
    = deg D
      - rank[H^0(C, K_C - B) -> H^0(D, (K_C - B)|_D)].

The proof: tensor 0 -> O_C(-D) -> O_C -> O_D -> 0 by B, use
h^1(C, B) = h^0(C, K_C - B)^*, and dualize the restriction map.

Apply this with B = jH and D = 2Z (or D = Z when j = 0), and use the
adjoint-jets ranks:
  h^0(C, O_C(Z)) = h^0(C, O_C) + 6 - 6 = 1,
  h^0(C, H+2Z) = h^0(C, H) + 12 - 12 = 2,
  h^0(C, 2H+2Z) = h^0(C, 2H) + 12 - 12 = 3,
  h^0(C, 3H+2Z) = h^0(C, 3H) + 12 - 10 = 6,
using h^0(C, H) = 2 (from j = 1) and h^0(C, 2H) = 3 (from j = 2).
The natural inclusions H^0(C, H) -> H^0(C, L) -> H^0(C, H+2Z)
agree on dimensions 2, 2, 2, so both are isomorphisms. In particular
H^0(C, L) = xi H^0(C, H), and h^0(C, L) = 2. Similarly H^0(C,
theta) = xi^2 H^0(C, 2H), of dimension 3. The pencil u is base-point
free by construction, so the common zeros of xi u equal div(xi) = Z.

For (4), the defining-divisor sequence for N = (k+1)L followed by
Serre duality shows that evaluation on Z is surjective if and only if
the natural inclusion H^0(C, (3-k)L) -> H^0(C, (3-k)L + Z) is an
isomorphism. For k = 2 this is L -> H+2Z, between spaces of dimension
2; for k = 3 it is O_C -> O_C(Z), between spaces of dimension 1; for
k >= 4 both spaces vanish because the target bundle has degree
10(3-k) + 6 <= -4. Item (5) restates the six-vector lemma and the
choice of Z. []

Riemann-Roch and Serre duality give two further identities on C:
  h^1(C, L) = h^0(C, 3L) = 12,   h^0(C, 4L) = 21.
Indeed, h^0(C, L) - h^1(C, L) = 10 + 1 - 21 = -10 and K_C - L = 3L,
so 2 - h^0(C, 3L) = -10; and h^0(C, 4L) = h^0(C, K_C) = g(C) = 21.

Finally, in compatible frames r = z u at each marked point, so
r'(q_j) = u(q_j) up to the frame. The six rows r'(q_j) in C^2 span
C^2: their projective directions are the marked values of f, and
f(q_j) are not all equal.