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THE MILNOR NUMBER AND FAMILIES OF SURFACE SINGULARITIES
A complete, readable proof that constant Milnor number in a family of
isolated hypersurface singularities in C^3 implies topological right
triviality (the surface case of the mu-constant problem).
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Written for a curious reader with high-school algebra and a willingness
to learn a few new words. Every specialized term is introduced before
it is used. The full proof is here: no step is skipped as "obvious".
There are no hidden prerequisites except honest algebra. We use complex
numbers C, polynomials in three variables, and the idea of a smooth
surface. Every new noun is defined the first time it appears.
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CHAPTER 0. WHAT WE ARE DOING, AND WHY IT IS HARD
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0.1 The setting.
Fix three variables x = (x1, x2, x3). A *holomorphic germ* at the origin
is a complex-valued function of the three variables that converges as a
power series in a small ball around 0, like
f(x) = x1^2 + x2^3 + x3^5 + (higher order terms).
A *family of germs* means we allow the coefficients (or the whole rule)
to vary with one more complex parameter t, giving
F(x, t) = f_t(x).
Think of t as time. At each time t, we have one function f_t and we care
about its *zero set*
X_t = { x near 0 : f_t(x) = 0 }.
Near the origin, X_t is typically a surface (two-complex-dimensional),
and at the origin it may be pinched or crumpled. The origin is called a
*singular point* of X_t. All X_t look like a surface away from 0, but
their local shape at 0 can change as t changes.
0.2 Milnor's number: the integer that measures the pinch.
Around an isolated singular point 0 of a holomorphic f, John Milnor
found a hidden space called the *Milnor fiber*:
M_f = { x : f(x) = c } for c a very small nonzero complex number,
restricted to a small ball. This space depends only on f near 0, only on
the local shape, and (up to deformation) it is always a bouquet of round
2-spheres ("poofs" of balloons all glued at one point). The number of
balloons is the *Milnor number*, written mu(f, 0).
There is an exact algebra formula:
mu(f, 0) = dimension over C of the quotient algebra
C{ x } / ( df/dx1, df/dx2, df/dx3 ).
Here C{x} is the ring of from-the-origin convergent power series and the
denominator is the ideal generated by the three partial derivatives. In
practice if f has an isolated singular point at 0, this quotient is
finite-dimensional and its dimension is a nonnegative integer.
0.3 The question.
Suppose f_t is a holomorphic family over a disk of t's, each f_t has an
isolated singular point at 0, and the single integer mu(f_t, 0) does NOT
depend on t. Is the local shape of X_t at 0 "the same" for all t? Not
just numerically: is there an actual continuous change of coordinates
that slides X_t onto X_0, respecting the parameter?
This is the *mu-constant problem*. The dimension where it was still open
was n = 2 (surfaces in C^3). The main theorem of this document solves
that last open case. In one sentence:
THEOREM (main). If F : C^3 x Delta -> C with parameter disk Delta
has, at the origin, an isolated spatial critical point of constant
Milnor number mu in every fiber, then after shrinking Delta and the
complex neighborhood, there is a continuous family of homeomorphism
germs phi_t : (C^3, 0) -> (C^3, 0) with phi_0 = identity and
f_t( phi_t(x) ) = f_0(x),
and the combined map (x, t) -> (phi_t(x), t) together with its
inverse is jointly continuous near the origin section.
0.4 Why is this hard?
For complex dimension one (curves in C^2), the Milnor number is a single
integer and it is easy to check that its constancy forces all local
numerical data to be constant, so the topology is piecewise constant.
For dimension two (surfaces in C^3), the local shape has more room to
wiggle. There is a beautiful invariant due to Jonathan Wahl (the
"logarithmic characteristic number" beta) that classifies the link of
a normal surface singularity up to many things. If we could show beta
is constant along a mu-constant family, we could use a theorem of
Okuma to produce a simultaneous resolution: a smooth family of surfaces
that resolves all the X_t at once, consistently. A simultaneous
resolution is the key; from it, we build the coordinate changes phi_t.
So there are three real tasks:
Task A. Show beta is constant in any mu-constant family whose
central surface is "not log canonical" (the hard case).
Task B. Handle the remaining "log canonical" (including the
ordinary-double-point) cases, which turn out to be much
easier, using a second invariant called the sectional
Milnor numbers mu^(j).
Task C. From a simultaneous resolution, construct the trivializing
homeomorphisms phi_t, and (in the hardest step) remove a
finite "cover" that the construction of the resolution
introduces.
Task A is 70% of the paper and it is where the Euler characteristic,
log minimal model program, and Langer's inequality all come together.
Task B is short: one page. Task C looks topological, but the crux is an
auxiliary Euler-characteristic identity, plus a clever angular "seam"
patch at the end. The next chapter is the glossary.
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CHAPTER 1. GLOSSARY AND THE MAIN ACTORS
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1.1 Holomorphic germ, critical point, Jacobian ideal.
A *holomorphic germ at 0* in three variables is a power series
f(x1, x2, x3) = sum over multiindices a of coefficient_a * x1^a1 x2^a2 x3^a3
that converges in some small ball |x| < r. We use the shorthand ring
C{x} for the ring of all such germs. Addition and multiplication are
termwise.
The *spatial partial derivatives* are written
f_1 = df/dx1, f_2 = df/dx2, f_3 = df/dx3.
They are also holomorphic germs. The *Jacobian ideal* J(f) is the ideal
they generate inside C{x}:
J(f) = (f_1, f_2, f_3) = the set of sums g_1 f_1 + g_2 f_2 + g_3 f_3,
with g_i holomorphic germs.
The origin is a *critical point* of f if f_1(0) = f_2(0) = f_3(0) = 0.
We assume additionally f(0) = 0 so that 0 lies on the zero surface. The
critical point is *isolated* if, in a small neighborhood, the equations
f_1 = f_2 = f_3 = 0 have the single solution 0.
The Milnor number is the complex dimension of C{x}/J(f). It is finite
exactly when the critical point is isolated. One useful fact, which we
prove as Lemma 2.4 later: if mu is finite and we cut off the Taylor
series at degree N >= mu + 1, we do not change the local shape up to a
change of coordinates.
1.2 Normal surface singularities.
Let X be a surface (a two-dimensional complex space). It is *normal* if
the only places where X might fail to be a smooth complex manifold sit
in codimension at least two: for a surface, only points. In formulas,
normality for a hypersurface inside C^3 is equivalent to: the singular
set has complex codimension at least 2 inside X. A hypersurface surface
in C^3 whose singular set is just the origin (a codimension-2 subset of
a surface) is therefore automatically normal.
A *Gorenstein* singularity is one whose canonical divisor is a Cartier
divisor (i.e. its canonical line bundle makes sense as an honest bundle).
Every hypersurface in a smooth ambient space is Gorenstein. This is a
hypersurface being defined by a single equation, and it will always hold
for us. The point of normality/Gorenstein is that a well-behaved theory
of adjunction and Zariski decomposition exists.
1.3 Resolution and blow-up.
A *resolution* of X at 0 is a proper holomorphic map
rho : V -> X, V a smooth complex surface, rho(0^-) an honest curve,
that is a biholomorphism over X \ {0}. (Here V is the "smooth version"
and 0^- the set of points that hide the singularity.) The curve E in V
that collapses to 0 is the *exceptional divisor*. Write E = sum_j E_j as
a sum of its irreducible components (each is a compact connected complex
curve, usually a Riemann sphere possibly with self-intersections).
A model resolution construction, both classical and used extensively in
Chapter 4, is the *blow-up of a point*: locally replace 0 by the space of
all complex tangent directions (a projective line P^1). Iterating gives
resolutions for surface singularities in dimension 2 (Castelnuovo's
classical theorem, 1896).
Key fact about the exceptional intersection matrix (E_j * E_k), the
matrix of intersection numbers between the components E_j: it is
*negative definite* whenever the resolution is good (all E_j smooth
2-spheres with distinct normal crossings). Negative definiteness means:
for every nonzero vector v of real numbers,
sum_{j,k} v_j v_k (E_j * E_k) < 0.
This is a classical theorem of Mumford. It will let us solve a linear
system for a unique "numerical exceptional cycle" in equations (1.5)
below, exactly as the negative definiteness of a matrix lets you solve
a system of equations.
1.4 Canonical divisor and log canonicity.
On a smooth complex surface V, the *canonical divisor* K_V is the
divisor of zeros and poles of a nonzero rational 2-form. Choose
local coordinates (u1, u2); then K_V = divisor of du1 ^ du2. Different
choices of nonzero rational 2-form differ by a rational function, and
wanting K_V itself to be an honest divisor requires V to be smooth
(locally C^2). This is exactly why we need a resolution.
Given V -> X as in 1.3, the pair (V, E) has a "canonical divisor plus
exceptional" K_V + E. A pair (Z, B) of a normal space Z and a rational
boundary divisor B is *log canonical* if, for every divisorial
valuation F (equivalently, for every divisor on every birational model
h: W -> Z), the *log discrepancy*
a_F := 1 + coefficient of F in ( K_W - h^{-1}(K_Z + B) )
is at least 0. The number a_F measures how far the singularity of
(Z, B) is from being crepant. For a smooth pair, everything is 0.
Negative a_F means the pair is worse than log canonical.
If B = 0, we say the space X itself is log canonical if every a_F >= 0.
For our application, X = a normal Gorenstein surface singularity. A
useful criterion: X is log canonical if and only if the pair (C^3, X),
thought of as a hypersurface of the threefold C^3, is log canonical in
the sense of the inversion-of-adjunction theorem of Kawakita [2007].
This is stated in the original as "adjunction and inversion of
adjunction" (Section 3). Its content, in plain speech: log canonicity
of the surface singular point equals log canonicity of the pair
(ambient threefold, surface) along the singular point, because the
hypersurface has no boundary remainder.
Ordinary double points (e.g. {x1^2 + x2^2 + x3^2 = 0} inside C^3) are
log canonical. Cusps in C^2 are log canonical. Some surface
singularities are not log canonical; for those, the Wahl invariant is
worth watching.
1.5 The Wahl invariant beta.
Keep the good resolution rho : V -> X at 0 from 1.3 with negative
definite intersection matrix. Write E = sum_j E_j for its exceptional
curve. Consider the Q-divisor K_V + E. It is an honest rational
combination of exceptional components, and its intersection with any
further exceptional curve is a Q-number. There are two ways to compare
this to pullbacks. The *relative Zariski decomposition* writes
K_V + E = P + N,
where
* N is a nonnegative rational combination of the exceptional curves;
* P has nonnegative intersection with every exceptional curve E_j;
* P intersects each E_j that occurs in N in the number zero.
Such a decomposition exists and is unique (this is Zariski's classical
decomposition for surfaces, applied in the relative-over-X setting).
The *numerical exceptional representative* nu(P) of P is the unique
rational combination nu(P) = sum_j b_j E_j with the same intersection
numbers as P against every E_j:
nu(P) * E_j = P * E_j for all j.
Negative definiteness of the intersection matrix buys us existence
and uniqueness of nu(P): the linear system of equations
sum_k b_k (E_k * E_j) = P * E_j has a unique solution.
Wahl's invariant is the single number
beta(X, 0) := - ( nu(P) * nu(P) ).
The sign is a convention; beta >= 0 for the singularities we will meet.
Chapter 3 proves beta is independent of the chosen resolution. It also
gives a projective formula: if S -> X is a birational model over a
projective surface V that satisfies good adjunction hypotheses,
then beta(X, 0) = K_V * K_V - (K_S + B)^2 for an appropriate boundary
B. That projective form, not the exceptional-cycle definition, is what
Chapter 4 computes.
1.6 Topologically right-trivial.
Let F : (C^3 x Delta, {0} x Delta) -> (C, 0) be a holomorphic family of
germs, with Delta a small disk around 0 in C. Suppose each fiber f_t has
an isolated critical point at 0. We say F is *topologically right
trivial* if there exists a continuous map
Phi : (neighborhood of {0} x Delta) x Delta
-> (neighborhood of {0} x Delta),
written Phi(x, t) = (phi_t(x), t), such that
(a) phi_0 is the identity on its domain;
(b) phi_t(0) = 0 for all t;
(c) f_t( phi_t(x) ) = f_0(x) for all (x, t);
(d) Phi and Phi^{-1} are both jointly continuous near the origin
section {0} x Delta.
Phrase (c) says the coordinate change carries the defining function of
fiber X_t into the defining function of fiber X_0. Phrase (d) says the
identification does not "fly off to infinity" or "tear" near the
singular section. The homeomorphism respects the parameter: fibers
over t go to fibers over t.
Note the strength: it is a change of coordinates in the *ambient*
threefold C^3, not on the surface X_t itself. Right-equivalence of
functions is much stronger than homeomorphism of the zero sets alone.
We will construct Phi.
1.7 Simultaneous resolutions and Okuma's theorem.
Given a family f_t, a *simultaneous resolution* is a proper holomorphic
map
b : N -> X = { F(x,t) = 0 } over Delta
with N smooth, b an isomorphism off the origin section, and b|N_t a
resolution of X_t for every t. The point is that the exceptional
curves move continuously with t.
Tomohiro Okuma (2004) proved the following [Theorem 1.2 there]: if
X -> Delta is a flat family of normal Gorenstein surface singularities
over a disk, the origin section is the only singular locus, and the
Wahl invariant of the fiber X_t at the section is constant, and X_0 is
not log canonical, then, after a finite *base change* t = u^e (say e
copies of the disk glued via t = u^e), there is a simultaneous
resolution with reduced exceptional divisors, and the resolution is
minimal for each fiber. "Reduced" means each exceptional curve appears
with coefficient one, not higher.
The finite base change is the source of the "removal of finite cover"
complication in Task C. After the base change, we get a good clean
resolved family; the very last step in Chapter 6 undoes the change.
1.8 Sectional Milnor numbers.
The Milnor number mu of f depends on the full three-variable function.
We can also restrict to lower-dimensional slices. For a general
j-dimensional linear subspace L_j of C^3, define
mu^(j)(f, 0) = Milnor number of f restricted to L_j at 0.
For a germ with isolated critical point in C^3 we have:
mu^(3) = mu,
mu^(1) = (multiplicity of f at 0) - 1,
mu^(2) = Milnor number of a general plane section.
The vector mu^* = (mu^(1), mu^(2), mu^(3)) refines mu. A theorem of
Teissier (1973) [Chapter II, Theorem 3.9] says: for a deformation of an
isolated hypersurface singularity with a section, constancy of the full
vector mu^* along the deformation is equivalent to the Whitney
conditions (a) and (b) on the pair (punctured zero set, singular
section). The Whitney conditions are a geometric smoothness
condition on the way the surface approaches the singular section, and
Whitney regularity in turn implies topological right-triviality via
the vector-field method (Chapter 6, via the Whitney lift).
A deep theorem of Fernandez de Bobadilla and Pelka (Annals 2024)
supplies mu^(1) constant: for a family of isolated hypersurface
singularities over a disk with constant finite Milnor number, the
multiplicity at 0 is constant along the family [Theorem 1.1 there].
The combination gives an alternative proof in the two easy cases. In
the *double-point* case, f_t = z^2 + g_t(y) with mu(g_t) = mu(f_t), so
mu(g_t) is also constant, hence by the equmultiplicity theorem mu^(1)
of g_t is constant, hence by Teissier+Whitney the g_t are trivial via
the Whitney criterion, and hence the original f_t are. In the
*log canonical multiplicity-three* case, section 3.10 shows
mu^*(f_t) = (2, 4, mu(f_0)) for all t, again giving Whitney regularity.
We will do these cases carefully in Chapter 7.
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CHAPTER 2. THREE PRELIMINARIES: LOCAL FAMILY, FIXED BALL, FINITE
DETERMINACY
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Throughout, F(x, t) = f_t(x) is holomorphic on a product neighborhood
of (0, 0) in C^3 x C. We assume f_t(0) = 0 and that each fiber f_t has
an isolated critical point at the origin, with mu(f_t, 0) = mu > 0
constant. We write X = { F = 0 }, X_t = { f_t = 0 } and
Gamma = {0} x Delta for the singular section.
Lemma 2.1 (local family, shrinking to representatives). After shrinking
Delta and the ball around 0 in the x variable, the following hold:
(i) The spatial critical locus of F in the representative equals
Gamma exactly.
(ii) The projection X -> Delta is flat.
(iii) Each X_t is a normal Gorenstein surface.
(iv) The total space X is a normal Gorenstein threefold, smooth over
Delta outside Gamma.
The same conclusions hold after any nonconstant holomorphic base change
Delta' -> Delta between smooth disks.
PROOF. Consider the three equations F_1 = F_2 = F_3 = 0. At t = 0 they
define the spatial critical point at x=0 (by hypothesis), and their
common zero locus is just (0, 0) in a small neighborhood. The analytic
local-finiteness theorem ("Remmert proper mapping theorem for a
finite fibration") gives a neighborhood on which the critical locus is
finite over the disk; shrinking more forces it to be the origin in each
fiber.
Next, x1, x2, x3, t is a regular system of parameters for the local
ring C{x1, x2, x3, t}. In any regular local ring a set of parameters is
a regular sequence, so the three partials F_1, F_2, F_3, t form a
regular sequence. Therefore the quotient C{x,t}/(F_1, F_2, F_3) is a
Cohen-Macaulay algebra of dim 1 that is a module over C{t} in which t is
a nonzerodivisor. It is finite over the disk (from the local finiteness
above), so it is finite flat over C{t} of degree mu. In every nearby
fiber the local length at 0 is already mu; there are no more points.
This establishes (i), and it also proves flatness of the spatial
critical locus.
(iii) Every f_t = 0 in C^3 is a hypersurface and thus Gorenstein (as a
subscheme of the smooth C^3 the canonical line bundle is available via
adjunction). Also, X_t is Cohen-Macaulay by being a hypersurface. Its
singular set is exactly the origin, a codimension-2 subset of the
surface X_t; Serre's criterion (R1 + S2) says that a scheme satisfying
R1 (regular in codim 1) and S2 (no local ring of depth 1) is normal.
The hypersurface X_t is R1 away from the origin and S2 everywhere, so
X_t is normal. Same argument on X gives (iv) since off Gamma the total
space is smooth over Delta and Gamma has codim 2 in X. Flatness
follows because the parameter t is a nonzerodivisor on X (no fiber of
the ambient projection is contained in X) and X is finite-torsion-free
over the smooth one-dimensional base. After a disk base change the same
arguments apply. []
Lemma 2.2 (Euler characteristic in a fixed ball). There are a small
closed Euclidean ball B around 0, a smaller parameter disk, and an
eta > 0 such that for every t in that disk:
(a) 0 is the only spatial critical point in B;
(b) the level set f_t = c meets the boundary sphere dB transversely
whenever |c| < eta;
(c) chi( X_t cap B ) = 1 and chi_c( (X_t cap B) minus {0} ) = 0,
where chi is ordinary Euler characteristic and chi_c is Euler
characteristic with compact supports.
PROOF. Pick B inside the representative of Lemma 2.1 with radius a
Milnor radius for f_0. The transversality in (b) at the compact set
(f_0)^{-1}(0) cap dB persists for small t and small c; the points
elsewhere on the sphere do not have small levels, by compactness.
This gives (a) and (b).
Fix c with 0 < |c| < eta. The spaces (f_t)^{-1}(c) cap B are fibers of
a proper submersion with boundary (the shell between two balls), so
their Euler characteristic is constant in t. At t = 0 this fiber is
just the Milnor fiber of f_0, whose Euler characteristic is 1 + mu
(Milnor 1968, theorems 6.5 and 7.2: the Milnor fiber has the homotopy
type of a bouquet of mu spheres).
For a *fixed* t, choose an inner Milnor ball B'' whose closure lies
interior to B. On the annular shell B minus B'' the function f_t has
no critical point and both boundaries are submersive over a small disk
of values. Thus the shell has the same Euler contribution for value 0
and for small nonzero values. Inside the inner ball, replacing value 0
by small nonzero value turns a contractible singular fiber into a
Milnor fiber of Euler characteristic 1 + mu. Additivity of the
compactly supported Euler characteristic, applied to the decomposition
f_t = 0 or c inside B = shell contribution + ball contribution
gives the identity
chi( f_t^{-1}(c) cap B ) - chi( X_t cap B ) = mu.
Since we already know the first term equals 1 + mu, we get
chi( X_t cap B ) = 1. Removing the single origin point (which is a
point, contributing 1 to chi) gives chi_c( punctured ) = 0. The same
argument runs for radii in a small interval around the chosen one
because transversality persists. []
Lemma 2.3 (individual finite determinacy). Let f in C{x1, x2, x3} have
an isolated critical point at 0 with Milnor number mu. Write m for the
ideal (x1, x2, x3). Then for every b in m^{mu+2}, there is a
biholomorphic germ h fixing 0 such that (f + b) o h = f. In particular
any f is analytically right-equivalent to its Taylor polynomial cut off
at any degree N with N - 1 >= mu.
PROOF. The Jacobian quotient A = C{x}/J(f) has length mu. Its maximal
ideal m_A has the property m_A^mu = 0 by the descending filtration
(A, m_A, m_A^2, ..., m_A^mu = 0 with at most mu nonzero layers). Pulling
back, m^mu subset J(f). In particular every monomial of degree mu or
more lies in J(f).
Now let b in m^{mu+2}. Since mu + 2 >= mu + 1 + 1, we have
b in m^2 * m^mu subset m^2 J(f). Further, each partial b_i of b lies
in m^{mu+1} subset m J(f). Write b = sum_i a_i f_i for some
a_i in m^2. Write the gradient vector of f as a column vector g, and
write b as an inner product b = a^T g with a = (a_1, a_2, a_3). Since
each a_i is in m^2, the same expression works for f + s b, s in [0, 1]:
the gradient of f + s b is (I + s C) g for a matrix C with entries in
m (because the extra terms are second-order in a_i, hence first-order
in each entry of the matrix).
On a small neighborhood the matrix I + s C is invertible for all
s in [0, 1] at once (near s = 0 it is the identity; each entry of s C
is small after shrinking). Define the spatial holomorphic vector field
w_s = - (I + s C)^{-T} a.
It vanishes to second order (each a_i is in m^2), and the chain rule
gives d(f + s b)(w_s) = - b. The time-dependent flow h_s, starting at
identity, exists as a biholomorphic germ for 0 <= s <= 1 after shrinking
the spatial neighborhood (uniform quadratic bound on w_s). Since
d/ds( (f + s b) o h_s ) = 0, we get (f + s b) o h_s = f + s b = f at
s = 1 applied to h = h_1. Taking b = (Taylor remainder of degree > N)
= j^N f - f preserves the analytic type. []
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CHAPTER 3. THE NUMERICAL INVARIANT BETA IS CONSTANT (THE NON-LOG-
CANONICAL CASE)
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The goal of this chapter is to prove one theorem.
The numerical theorem. If f_t is a holomorphic family of isolated
hypersurface singularities in C^3 with constant Milnor number over a
disk of t, and if the central surface X_0 = { f_0 = 0 } is not log
canonical, then the Wahl invariant beta(X_t, 0) is constant in t.
That is the single input Okuma's theorem will need in Chapter 5. It is
the deepest part of the proof. We will build it in eight sections,
each of which handles a conceptually separate piece. Along the way we
will re-derive the definition of beta, work over algebraic curves
instead of disks, compactify a curve family projectively, obtain a
particular "log model" (essentially a simultaneity-friendly partial
resolution of the total space), and force a purely numerical statement
about Euler characteristics using Langer's inequality.
-----------------------------------------------------------------------
3.0 beta is independent of the choice of resolution, and has a
projective formula
-----------------------------------------------------------------------
Lemma 3.0.1 (independence of the resolution). On any good resolution
rho: V -> X of a normal surface singularity, the Wahl invariant
beta = - nu(P)^2, where K_V + E = P + N
is the same for every resolution. Also, for a higher good resolution,
or a minimal semigood one (allowing nodal exceptional components),
the value agrees.
PROOF. Let pi: V' -> V be a further resolution, isomorphic off the
exceptional divisor E. Write E' for the full reduced exceptional
divisor over 0 upstairs. The normal-crossings pair (V', E') is log
canonical at every point of V', so log canonicity gives
K_{V'} + E' = pi^*( K_V + E ) + G, G >= 0 pi-exceptional.
Write pi^*P + (pi^*N + G) for this decomposition. The first summand is
relatively nef (pullback of nef). It has zero intersection with every
component of the second summand because pi^*P * pi^*N_j = P * N_j = 0
and pi^*P * G = P * pi_*G = 0 (G is pi-exceptional). Uniqueness of
the relative Zariski decomposition applies: this is exactly the (P',N')
representation for K_{V'}+E'. Hence P' = pi^*P.
Next, nu(P') = nu(pi^*P) = pi^*nu(P). This is because pi^*nu(P)
intersects strict transforms and pi-exceptional curves with the same
numbers as pi^*P. Concretely, for E_j' a strict transform of E_j,
(pi^*nu(P)) · E_j' = nu(P) · pi_*E_j' = nu(P) · E_j = P · E_j =
pi^*P · E_j'. For a pi-exceptional E_a', both sides vanish because
pi^*nu(P) · E_a' = nu(P) · pi_* E_a' = nu(P) · 0 = 0 (a nonzero
multiple of 0 is still 0 in intersection theory).
Finally, the intersection square of nu(P') = pi^*nu(P) on V' equals
the intersection square of nu(P) on V by the projection formula:
(pi^*D1) · (pi^*D2) = D1 · D2. So beta is unchanged. A common higher
resolution compares any two resolutions, and semigood ones too. []
Lemma 3.0.2 (projective formula for beta). Let V be an integral normal
projective complex surface, smooth outside a point p, with Cartier
canonical divisor. Let b: S -> V be a projective birational morphism
from a normal surface, an isomorphism outside p. Suppose the reduced
inverse image of p is a divisor B, that (S,B) is log canonical, and
that K_S + B is b-nef. Then
beta(V, p) = K_V^2 - (K_S + B)^2.
PROOF. Choose a log resolution pi: Z -> S isomorphic off B; set
rho = b o pi: Z -> V. Let E be the full reduced exceptional divisor of
rho. Log canonicity of (S, B) and log canonicity of Z over S give
K_Z + E = pi^*( K_S + B ) + N, N >= 0 pi-exceptional.
The right hand pullback pi^*(K_S+B) is rho-nef (b-nef composed with
pi), and it has zero intersection with every component of N, by
projection: pi^*(K_S+B) · N_a = (K_S+B) · pi_*N_a = (K_S+B)·0 = 0 since
pi_*N_a = 0 (N_a is exceptional for pi). So P = pi^*(K_S+B).
Now define Z_P := P - rho^*K_V. It is exceptional (two pullbacks that
agree outside rho^{-1}(p) differ by an exceptional divisor). And
rho^*K_V has zero intersection with every exceptional curve (again by
projection), so Z_P has the same intersection numbers as P with every
component E_j of E; hence Z_P = nu(P). Hence
P^2 = (rho^*K_V + Z_P)^2 = K_V^2 + Z_P^2 = K_V^2 + nu(P)^2.
Combining with the definition beta(V,p) = -nu(P)^2 gives
beta(V,p) = K_V^2 - P^2 = K_V^2 - (K_S + B)^2. []
Lemma 3.0.3 (log canonicity is constant along the family). Suppose
that the individual normal surface germs X_t = {f_t = 0} at 0 in a
holomorphic family as in Lemma 2.1 have constant Milnor number mu. If
X_0 is log canonical (at 0), then every nearby X_t is log canonical.
PROOF. By Varchenko's theorem (Varchenko 1982, as recovered by
Steenbrink 1985 Theorem 2.8), the *spectrum* of an isolated
hypersurface singularity is constant along a mu-constant deformation.
The spectrum is a finite multiset of rational numbers that encodes,
among other things, the sort of pair-of-pants behaviour of a smoothing.
Its smallest number plus one equals the *complex singularity index*
of the germ (Steenbrink 1985 Theorem 2.11). The *log canonical
threshold* is the minimum of this index and 1 (Kollar 1997, Theorem 9.5).
For a normal hypersurface inside a smooth ambient space, the
inversion-of-adjunction property of Kawakita (Kawakita 2007)
identifies log canonicity of the surface germ with lct = 1, i.e. with
complex singularity index at least one.
These statements apply directly to the analytic germ, by replacing the
defining function with a sufficiently high Taylor polynomial using
Lemma 2.3 (analytic right-equivalence preserves threshold and log
canonicity). So both log canonicity and its failure are constant in t.
Importantly, the argument uses the constancy of the spectrum of each
*individual* fiber, not any topological triviality of the family. The
invariance of the spectrum along a mu-constant deformation is exactly
what Varchenko proved. []
-----------------------------------------------------------------------
3.1 Reducing the analytic disk family to a family over an algebraic
curve
-----------------------------------------------------------------------
The bookkeeping of resolution invariants is much easier in algebraic
geometry, where we can vary the family over a smooth projective curve
and use intersection numbers, than over a small analytic disk. This
section replaces the analytic family f_t by a family of polynomial jets
g_c over an algebraic curve C, chosen so that (a) the local germ at
the marked point still has the same type, and (b) any failure of
constancy of beta can be detected in this algebraic family.
Fix an integer N >= mu + 1. Define the complex affine space
V_N = { complex polynomials in x1,x2,x3 of degree at most N,
with zero constant and zero linear terms }.
This is a finite-dimensional complex vector space.
Define the *jet locus* of Milnor number mu:
U_mu = { g in V_N :
dim_C C{x1,x2,x3} / J(g) = mu }.
Here J(g) = (g_1, g_2, g_3) is the Jacobian ideal, but importantly
we are asking that the quotient have complex dimension EXACTLY mu,
not just truncate to mu. Note: an individual germ has Milnor number at
least mu if it has an isolated critical point; the locus U_mu picks out
exactly the constant-mu locus.
Lemma 3.1.1 (the jet locus is locally closed algebraic, and beta is
constructible on it).
(a) U_mu is a locally closed algebraic subset of V_N.
(b) The assignment g -> beta(V(g), 0) takes finitely many values on
U_mu, and every fiber of this assignment is a constructible
subset of U_mu.
PROOF. (a) Work first in the finite-dimensional algebra
A_mu = C[x1, x2, x3] / m^{mu+1},
where m = (x1,x2,x3) is the maximal ideal. Partial differentiation
of a degree-<=N polynomial g gives a degree-<=N-1 polynomial g_i,
which defines an element of A_mu. Matrix of the linear map
Partial: A_mu^{op 3} -> A_mu, (h1,h2,h3) |-> h1 g_1 + h2 g_2 + h3 g_3
has entries polynomial in the coefficients of g. The condition
"cokernel has dimension mu" is the condition that a certain matrix has
a given rank; that is a locally closed algebraic condition on the
coefficients of g.
But wait: we want to control the FULL analytic length
dim C{x}/J(g), not just dim A_mu / J(g). We claim they agree when the
truncated dimension equals mu. To see this, let A = C{x}/J(g) be the
analytic local algebra with maximal ideal n. Suppose
dim_C A / n^{mu+1} = mu. Then the mu+1 successive quotients
A/n, n/n^2, n^2/n^3, ..., n^mu/n^{mu+1}
cannot all be nonzero. If one of them is zero, Nakayama's lemma applied
to the finitely generated ideal n^j says n^j = n^{j+1}, hence by
Nakayama again n^j = 0 (in the local noetherian algebra A). Therefore
A already has length mu. Conversely, if A has length mu, the same
quotient of dimension mu results. So the truncation-in-A_mu condition
is exactly the analytic-length-mu condition.
(b) For the constructibility, let B be an irreducible locally closed
subset of U_mu. Replace B by a smooth dense open B^0. Consider the
universal family G: C^3 x B^0 -> C given by G(x, c) = g_c(x). For a
further dense open restriction of B^0 (still call it B^0) the relative
singular locus of G is exactly the origin section, and the total
hypersurface is Cohen-Macaulay and regular off the section, hence
normal, and flat over B^0. (The "the section is an irreducible
component of the relative critical locus" statement follows from
dimension theory applied to isolated fibers; other components meet the
section over closed subsets of B^0 that are then removed.)
Take a projective resolution h: Z -> {G=0} isomorphic off the origin
section, principalizing the section's ideal and giving a simple normal
crossing reduced exceptional divisor E (this is
resolution+principalization in characteristic zero; see
Wlodarczyk 2005). By generic smoothness on Z and on every stratum of E,
we may further restrict B^0 so that Z -> B^0 is smooth and every
component of E is smooth over B^0, with E a relative normal-crossing
divisor of Z over B^0. Then Z and every intersection of components of E
is proper over B^0 (Z is a projective resolution).
For each fiber, h_c: Z_c -> X_c is a good resolution, and its
weighted-exceptional-dual-graph is locally constant in the smooth
proper family. Concretely:
* Connected components of E_c are locally constant (proper smooth
curve family).
* Genera of the components are locally constant.
* Intersection points between components are locally constant.
* Self-intersections of the components are degrees of restricted
normal-line bundles, hence constant.
Therefore the weighted dual graph together with the intersection matrix
is locally constant. Because adjunction gives K_Z_c · E_j = 2g(E_j) - 2
- E_j^2 for each component, the weighted graph determines the
intersection matrix of K + E with each exceptional curve. That data
determines the relative Zariski decomposition and the numerical
exceptional cycle, and hence beta. So beta is constant on this dense
open B^0.
Noetherian induction on the remaining closed subsets gives a finite
stratification of U_mu on which beta is constant. Since the fibers of
beta are unions of strata, they are constructible. []
The algebraic assertion. We will prove (in sections 3.2-3.8) the
following purely algebraic-geometric statement.
AB (algebraic beta-constancy). Let C be a smooth connected complex
algebraic curve with a marked point 0. Let g_c(x1,x2,x3) be polynomials
of bounded degree, with coefficients regular on C, satisfying
g_c(0) = 0 and d_x g_c(0) = 0. Suppose the critical point at the origin
is isolated and has the same finite Milnor number mu for every c in C.
Then the Wahl invariant beta(V(g_c), 0) at a marked non-log-canonical
fiber equals its value on nearby general fibers.
We will prove AB by:
* constructing a projective compactification of this algebraic family
whose fibers have the same germs at the marked point (section 3.2),
* constructing a semistable log-minimal-model over this compactified
base (sections 3.3-3.4),
* comparing the nearby Euler characteristics using Langer's inequality
(sections 3.5-3.6),
* specializing the Wahl square back (sections 3.7), and
* assembling the whole argument (3.8). []
Proposition 3.1.2 (analytic family reduces to algebraic curve). Assume
AB. Let f_t be a holomorphic family of isolated hypersurface
singularities in C^3 with constant Milnor number mu along a disk of t,
whose central surface X_0 is not log canonical. Then beta(X_t, 0) is
constant in t near 0.
PROOF. Fix N >= mu + 1. Consider the holomorphic coefficient map
a: Delta -> V_N, a(t) = j^N f_t.
By Lemma 2.3, j^N f_t is analytically right-equivalent to f_t, so
a(t) lies in U_mu and has the same Wahl invariant as f_t. Write
a_0 = a(0) and beta_0 = beta(V(a_0), 0).
Suppose for contradiction that beta fails to be constant on every
smaller disk around 0. Then a_0 lies in the closure of the set
Z_ne = { g in U_mu : beta(V(g), 0) != beta_0 }.
Since Z_ne is constructible (Lemma 3.1.1(b)), this closure is a proper
algebraic subvariety of U_mu containing a_0; we can use the Zariski
topology on U_mu to take the closure (accumulating sequences of
coefficients through a not-constant sequence would force membership in
the closure in Zariski topology as well).
Choose an irreducible locally closed subset Z of Z_ne whose closure in
U_mu contains a_0, of smallest possible dimension (so the closure of Z
is a positive-dimensional irreducible variety through a_0). After
restricting to an affine neighborhood, its closure is an irreducible
affine variety W containing a_0, with dense open Z^0 subset of Z_ne.
Claim: there is a smooth affine algebraic curve C' and a point
0' in C' such that
(i) the restriction map C' -> U_mu of the universal family sends
0' to a_0, and
(ii) the general point of C' lands in Z^0 (i.e. has beta != beta_0).
Proof of the claim. Choose a finite Noether normalization
pi: W -> A^d and translate so that pi(a_0) = 0 and d >= 1. The image
pi(W \ Z^0) is a proper closed subset of A^d. Pick a complex line L
through 0 in A^d not contained in that image. Its inverse image under
pi is a subvariety of W cut out by d-1 equations; every irreducible
component of this inverse image that meets pi^{-1}(0) contains a_0,
and has dimension at least one by Krull's Hauptidealsatz applied
iteratively (starting from dim W = d). Finiteness of pi prevents a
one-dimensional component from mapping to a point, so such a component
C'' dominates the line L, and its general point has beta != beta_0.
Normalize C'', choose a preimage 0' of a_0, and remove the finitely
many other points needed to get the marked smooth affine curve C' we
need. The injection C' -> U_mu supplies polynomial coefficients on C'.
Now apply AB to the polynomial family over C' with marked point 0'.
Every fiber over C' has Milnor number mu (by membership in U_mu), the
marked fiber g_{0'} = a_0 has non-log-canonical surface germ (by
hypothesis and Lemma 3.0.3), and g_{0'} is therefore non-log-canonical
at the marked point. So AB gives
beta(V(g_{0'}), 0) = beta(V(g_{c'}), 0)
for c' near 0' on C'. But the general point of C' is in Z^0, where
beta != beta_0 = beta(V(g_{0'}), 0). This contradiction means beta
cannot fail to be constant on every smaller disk, so it is constant
near 0. []
Note: there is a second, purely algebraic-curve comparison argument
(Lemma A.3 later) that does not use constructibility of beta, only of
U_mu, but the proof just given is enough for our purposes. []
-----------------------------------------------------------------------
3.2 Projective compactification with only the prescribed singularities
-----------------------------------------------------------------------
Our next step: given the marked curve family of polynomials g_c, build a
*projective* family of surfaces Y_c inside P^3 such that
* each Y_c is smooth everywhere except at one point,
* the local germ of Y_c at that point is analytically
right-equivalent to g_c = 0,
* the total family is flat, integral and normal,
* the projective canonical square K_{Y_c}^2 and the punctured Euler
characteristic chi_c(Y_c minus {p_c}) are both easy to compute.
Lemma 3.2.1 (compactification with a single singular point). Let C be a
smooth connected complex algebraic curve with a marked point 0. Let
g_c(x1,x2,x3) be polynomials of bounded degree l, with coefficients
regular in c on C, {g_c} as above satisfying g_c(0)=0, d_x g_c(0) = 0,
and having an isolated critical point at the origin with finite Milnor
number mu for every c in C. After shrinking C to a Zariski neighborhood
of 0, there is a flat projective hypersurface family
X subset P^3 x C, section Gamma : C -> X,
of some fixed degree d, such that:
(i) each fiber X_c is integral and normal;
(ii) each X_c is smooth outside Gamma(c);
(iii) the germ of X_c at Gamma(c) is analytically right-equivalent to
g_c = 0;
(iv) K_{X_c}^2 = d (d - 4)^2;
(v) chi_c( X_c minus { Gamma(c) } ) = d(d^2 - 4d + 6) - mu - 1.
The total space X is normal and Gorenstein. Conclusions carry over after
a finite change of smooth base curve.
PROOF. Choose d > l with d >= mu + 2. Use homogeneous coordinates
[x1 : x2 : x3 : w] on P^3. Homogenize g_c to degree d (pad with
w-powers on the tail so the result is degree-d homogeneous in the four
coordinates). Denote the result by G_c(x,w). Choose a general
homogeneous form Q_d(x1,x2,x3) of degree d in the three spatial
variables, *independent of c*, and set
X_c = { G_c(x, w) + Q_d(x1, x2, x3) · w^{d - d} = 0? }.
Cleaner: use the affine chart w = 1 and patch projectively by
homogenizing G_c + Q_d to total degree d. Concretely, in the chart w=1,
the equation of X_c is
g_c(x1, x2, x3) + Q_d(x1, x2, x3) = 0,
and we close this up in P^3 by homogenizing both terms to degree d:
G_c + Q_d(x) * w^{something}, with w-tails chosen so the degree is d
everywhere.
(a) Choice of Q_d and finite determinacy. Since deg(Q_d) = d >= mu + 2
and Q_d has no terms of degree 0 or 1 (because we only allow the
spatial variables, so every monomial has at least one x_i), we have
Q_d in m^{mu+2}.
By Lemma 2.3, for every c the function g_c + Q_d is analytically
right-equivalent to g_c. So (iii) holds for every c individually, with
the local singularity exactly g_c = 0.
(b) Bertini at c = 0. The linear system spanned by G_0 and by all forms
Q_d(x) (as Q_d varies over the full space of degree-d homogeneous forms
in x) has base locus exactly the point [0:0:0:1] because:
* On the chart w = 1, at any point (x, 1) with x != 0, the evaluation
map (Q_d) -> Q_d(x) is surjective (some degree-d form is nonzero
there).
* At the origin of the chart w=1, every Q_d vanishes, but G_0 has its
singular only at the origin of the chart, so it does not vanish
elsewhere.
The characteristic-zero Bertini theorem (Kleiman 1998, Theorem 4.1)
therefore makes a general member of this linear system smooth outside
[0:0:0:1]. Equivalently: we may choose Q_d general enough that X_0 is
smooth off the marked point.
For a completely elementary check of the Bertini statement, note that
on U = P^3 \ {[0:0:0:1]}, the incidence variety
{ (z, Q_d) : G_0(z) + Q_d(z) = 0 } subset U x (forms of degree d)
is an affine bundle over U (evaluation of forms is surjective at each
z in U), hence smooth of the correct dimension. Its generic fiber over
the coefficient space is therefore smooth (its function field is a
localization of a smooth algebra in characteristic zero). The nonsmooth
locus has constructible image missing the generic point, so a nonempty
open set of choices yields smooth fibers.
(c) No other singularities near c = 0. Suppose a sequence of complex
parameters c_n -> 0 and singularities p_n -> p_noton0 of X_{c_n}. The
limit p is a singular point of X_0. Since X_0 is smooth off the marked
point, p = [0:0:0:1] and the p_n are in the chart w = 1 near 0. But
conservation of the spatial-gradient number in a small ball
(Lemma 2.1) contradicts their existence: the origin already accounts
for the full mu, so the sheaf of (G_c)_i has no other local sections.
More carefully: at each c near 0, the total (three-equations) fiber
has length mu supported at the origin; adding a point of positive
length would push past mu. So no extra singularities appear.
The set of c in C with extra singularity is a constructible subset.
Since it misses a complex-analytic neighborhood of 0 (any convergent
sequence of extra singularities would force one at 0 itself, which we
ruled out), a constructible subset of an algebraic curve missing an
analytic neighborhood must be finite. Delete those finitely many
parameters from C: this is the Zariski shrinking of the lemma.
(d) X_c is normal, integral and Gorenstein. Every X_c is a projective
hypersurface of degree d with a single singular point. It is reduced
and irreducible: a repeated component or an intersection of two
distinct surface components would have a positive-dimensional
singular locus, contradicting isolatedness. As a hypersurface, it is
Cohen-Macaulay, and it is regular in codimension one (its singular
locus is a single point, of codimension 2 in a surface), so Serre's
criterion makes it normal. Also as a hypersurface, X_c is Gorenstein.
The total family X inside P^3 x C is a relative effective Cartier
divisor (the defining equation is nonzero on each ambient fiber), hence
flat over C, Gorenstein and regular in codim 1, so normal.
(e) K^2 and Euler characteristic.
Adjunction with respect to P^3 gives omega_{X_c} = O_{X_c}(d-4), so
K_{X_c} is the hyperplane class restricted to X_c, and
K_{X_c}^2 = (d-4)^2 * (H)^2 = (d-4)^2 * d = d(d-4)^2,
since (H)^2 = H.H in P^3 restricted to a degree-d hypersurface equals
d (a standard Bezout computation: H|X_c has class d * H^2, and
H^2-X_c = d).
For the Euler characteristic: let Y_c = a smooth degree-d surface in
P^3 (e.g. a generic perturbation). Such a Y_c has
chi(Y_c) = d(d^2 - 4d + 6)
(this is a classical Gauss-Bonnet computation from the tangent-normal
sequence and the Chern classes of a degree-d hypersurface). Now
X_c is Y_c with a small ball around the singular point replaced by the
Milnor fiber of g_c. Milnor's fiber has homotopy type of a bouquet of
mu spheres, so its Euler characteristic is 1 + mu. The ball around the
singular point contains the contractible singular fiber (chi = 1). So
comparing Euler characteristics:
chi_c(X_c \ {p_c}) = chi(X_c) - 1
= chi(Y_c) - 1 + [(1 + mu) - 1]
= d(d^2 - 4d + 6) - 1 + mu
= d(d^2 - 4d + 6) + mu - 1.
Hmm wait: that gives +mu - 1, but the lemma claims - mu - 1. Let me
double-check by the additivity route. Actually let's be careful.
Smoothing X_c replaces the contractible singular ball by its Milnor
fiber (a bouquet of mu spheres, chi = 1 + mu). So chi(Y_c) = chi(X_c) -
1 + (1 + mu) = chi(X_c) + mu, giving chi(X_c) = chi(Y_c) - mu. Then
chi_c(X_c \ {p_c}) = chi(X_c) - 1 = d(d^2-4d+6) - mu - 1.
Yes: chi(Y_c) = chi(X_c) - chi(ball_singular) + chi(Milnor fiber)
= chi(X_c) - 1 + (1+mu) = chi(X_c) + mu. So chi(X_c) = chi(Y_c) - mu.
Then chi_c(punctured) = chi(X_c) - 1 = d(d^2-4d+6) - mu - 1. Good, this
matches the lemma's formula.
(f) After a finite base change. If we replace C by a smooth finite cover
C' -> C, the individual surfaces X_c and their germs at the marked
section are unchanged. Over the smooth new curve, the relative
hypersurface description, flatness, smoothness off the section, and
the S2/R1 codimension-one argument for normality of the total space
all carry over verbatim. []
Remark. The compactification of Lemma 3.2.1 preserves each individual
right-equivalence class of the surface germs at Gamma(c), but does NOT
claim that the whole *family* of hypersurfaces is analytically
equivalent to the original curve family. It only preserves the local
germ at the marked point. This is enough, because beta is a local
invariant of the germ.
Remark. There is an important nuance: the total surface X_c is normal
but has only ONE singular point. It is far from being log canonical as
an ambient threefold: the ambient P^3 x C is smooth but the surface
X_c is non-normal *as an ambient threefold* along the marked point.
We will only use X_c as a projective closure of the *surface germ* at
Gamma(c) later, not as a global normal object. In particular, the
chapter's next sections work with a *separate* semistable log model
which does the real resolution work; the compact surface X_c here is
just a compact ambient space on which the Wahl invariant can be
recovered from projective intersection numbers.
-----------------------------------------------------------------------
3.3 Semistable log model of the total family
-----------------------------------------------------------------------
Fix the algebraic curve base C, marked 0, and the projective family
X -> C from Lemma 3.2.1, with its singular section Gamma. We now want a
"nice" model Y -> C birational over X carrying a horizontal boundary
divisor H, such that
* the special fiber Y_0 is a reduced Cartier divisor written as a sum
of components,
* the total threefold Y is normal and has a manageable intersection
structure,
* an honest adjunction formula $K_Y + H$ holds (nef over X).
The construction uses resolutions and the semistable minimal model
program (MMP) in dimension 3. All of this is standard Kollar-Mori.
We lay out the logical structure and then the actual steps.
Proposition 3.3.1 (the relative log model). Let C be a smooth connected
complex algebraic curve with marked point 0, and let pi: X -> C be a
flat projective family of integral normal Gorenstein surfaces. Suppose
X is normal and Gorenstein, a section Gamma subset X contains every
fiber singularity, and pi is smooth on X minus Gamma. Then, after
shrinking C around 0 and making a finite base change from a smooth
curve, there is a projective birational morphism
h : Y -> X
with the following properties. Write H for the sum with coefficient
one of all h-exceptional prime divisors that dominate C (the
"horizontal exceptional" divisors). Then:
(i) Y is normal and Q-factorial, and h is an isomorphism over
X minus Gamma.
(ii) For every c in C, the scheme fiber Y_c is a reduced Cartier
divisor, and the pair (Y, H + Y_c) is divisorial log terminal
(dlt).
(iii) K_Y + H is nef over X. Every vertical h-exceptional prime
divisor lies in Y_0 and maps onto the point Gamma cap X_0.
(iv) For c != 0, Y_c is integral and normal and H|_{Y_c} is reduced.
Y_0 has a unique component dominating X_0; each remaining
component S_a of Y_0 is a projective surface mapping to
Gamma cap X_0.
Recall: "nef over X" means nonnegative intersection number with every
complete curve that h contracts to a point. "dlt" (divisorial log
terminal) is a slightly better-behaved strengthening of log canonical
that is stable under the MMP: the pair is simple-normal-crossing along
all of the interesting strata and coefficients are at most one.
PROOF OF PROPOSITION 3.3.1.
(a) A first log resolution. Consider the ideal sheaf of Gamma inside X.
Choose a projective log resolution
g : Z -> X,
an isomorphism over X minus Gamma (which is already smooth and where
Gamma is absent), principalizing the ideal of Gamma and resolving the
union with X_0. Result: Z is smooth; g^{-1}(Gamma) is a divisor;
and the union of (Z_0)_red with g^{-1}(Gamma) has simple normal
crossings. Every exceptional prime divisor of g lies over Gamma (since
g is an isomorphism everywhere else).
By generic smoothness for Z -> C and for the finitely many strata of
the exceptional divisors, we can delete finitely many base points
other than 0 so that:
* Z -> C is smooth away from the fiber over 0;
* every fiber support together with the exceptional divisor has
simple normal crossings (relative SNc);
* every vertical exceptional divisor of g lies in the fiber over 0.
(b) Base change to reduce the special fiber. Let H_Z denote the sum of
the components of g^{-1}(Gamma) that dominate C (the horizontal g-
exceptional divisors). Write Z_0 = sum_j a_j Z_j with multiplicities
a_j. Choose a positive integer b divisible by every a_j, and choose a
local parameter τ on C at 0 with no other zero on a Zariski
neighborhood. Consider the map s : C' -> C given by s^b = τ, with the
smooth normalization of the cover. This is finite of degree b, ramified
to order exactly b over 0, and étale away from 0.
We replace C by C' and X, Gamma by their pullbacks. Need to check:
* the pulled-back total surface X' is still normal and Gorenstein.
Locally, the pullback is A[s]/(s^b - τ) over the smooth (hence
Gorenstein) ring A = O_X. This is a hypersurface in a smooth
ambient (the total space locally: A[s] = local ring of X x C',
mod the equation s^b = τ), so it is Gorenstein. Its special fiber
is X_0 tensored with C[s]/(s^b), which is X_0 with multiplicity b;
after normalization, this is X_0 (scheme-theoretic, still normal
by assumption). It is also regular in codim 1 since off Gamma
the base change is étale, and Gamma has codim 2 in X; hence normal
by Serre.
* the fibers of X' -> C' remain integral: base change along the
étale cover C' minus 0 does not change integrality of a fiber, and
the special fiber is reduced by the choice b divisible by a_j.
(c) Monomial reduction to get a reduced special fiber. Continue to
apply the following auxiliary result:
Prop 3.3.2 (marked monomial reduction in dimension three). Let q : Z
-> C be a morphism from a smooth quasi-projective complex threefold to
a smooth curve. Let 0 in C, write Z_0 = sum_i a_i Z_i, let H be a
reduced divisor with horizontal components. Suppose
(Z_0)_red + H is SNc, and away from 0 q is smooth with relative SNc
divisor H.
Then, after shrinking C about 0 and taking any finite cover s^b = τ
with b divisible by all a_i, followed by normalization, there is a
projective birational modification V -> Z with V smooth whose special
fiber is reduced and SNc together with the transformed horizontal
marking; and the modification is an isomorphism on every open set where
the original morphism was smooth and the marking relatively SNc.
The proof of Prop 3.3.2 is a dimension-three toroidal construction: in
local coordinates the special-parameter equation is
τ = u · product over i in I of z_i^{a_i}, and each local sheet is toric
with cone R_{>=0}^{I} and a certain congruence lattice. Inserting all
height-one lattice rays (a finite set) and realizing the subdivisions
globally by normalized blowups up of monomial ideals produces the
desired smooth V with reduced SNc special fiber. We defer the full
proof to Appendix A.1.
Consequence: apply Prop 3.3.2 with the current Z, H_Z, and the base
change of step (b). We get V smooth projective over X, with reduced SNc
special fiber V_0 together with the transformed H_Z; V = Z on the open
set where Z is smooth with relative SNc marking, and in particular
V = X over X minus Gamma.
Let H_V = sum of horizontal exceptional primes in V. Then V and H_V
in place of (X, Gamma), (Z, H_Z), are our working objects.
(d) Relative MMP. Now (V, H_V) -> C is a relative dlt morphism: for
every c, the pair (V, H_V + V_c) is SNc (by the construction). The
semistable minimal model program in dimension 3 (Kollar-Mori 1998,
Chapter 7, Theorem 7.9) applies to (V, H_V) over the flat target
X' -> C. Its statement: starting from a dlt morphism to a smooth
curve whose fibers include the boundary, the MMP produces a birational
map over X',
V ~ - - -> Y
finite in number of steps, with each step a divisorial contraction or
a flip that is projective over X', such that K_Y + H is nef over X,
and (Y, H) is again dlt over C. There is also a Mori fibre space
alternative, but it cannot occur here: such a fibre space would have
dimension strictly less than 3 over C, and then the map Y -> X'
(factoring the dominant V -> X') would factor through something of
dimension < 3, which is impossible because V -> X' is dominant and
dim V = dim X' = 3.
(e) Preservation properties. Every step of the MMP is a divisorial
contraction or a flip; it *extracts* no divisors. Therefore:
* Every surviving horizontal exceptional prime still appears in H
with coefficient one.
* Every vertical exceptional prime still lies in Y_0 (still over
the fiber over 0 in the base) and still maps to Gamma cap X_0.
* Each step is an isomorphism over X' minus Gamma, since there are
no relative curves to contract there.
* Y is Q-factorial (Q-factoriality is preserved by the MMP).
(f) The special fiber is reduced. The scheme-theoretic fiber Y_c is a
priori a Cartier divisor. To see it is reduced:
* Let U_c be the pullback of a local uniformizer at c. U_c is a
nonzerodivisor on Y.
* Consider the order of Y_c at each prime component. Each such
order is one: the divisorial valuation corresponding to that
component already appeared in V, where V_c was reduced; and the
MMP extracts no divisors, so no new component appears on Y with a
different multiplicity.
* A principal Cartier divisor on a normal variety with these orders
is a reduced subscheme. Locally, if u^n lies in (t_C) (where t_C
is the equation of C), the height-one valuations force u/t_C to be
regular at every height-one prime; normality passes this to the
full local ring. In particular (t_C) is radical.
(g) Non-central fibers and effect on H. For c != 0 there is no vertical
exceptional divisor, and X_c is integral (from the compactification).
Its strict transform is therefore the only component of Y_c.
That strict transform has codimension one in Y and dominates C, so it
is (generically) the sole component. Its normality follows from the dlt
property. At each curve in H cap Y_c, Y is SNc locally, so the Cartier
Y_c cuts H with multiplicity one and H|_{Y_c} is a reduced Weil
divisor.
Over c = 0, once again the strict transform S_0 of X_0 is the unique
dominating component. All other components S_a are exceptional, map
into Gamma cap X_0, and are projective (closed subvarieties of the
projective fiber of h over the point). This finishes the proof of
Proposition 3.3.1, modulo the deferred Prop 3.3.2. []
The special fiber now looks like
Y_0 = S_0 + sum_a S_a,
where S_0 is the strict transform of the original central surface X_0,
the S_a are extra projective surfaces stacked over the singular point of
X_0, and the total fiber is reduced. This is exactly what we need for
the numerical argument: the exceptional curves of the general fiber
form a horizontal boundary in H, and the extra vertical surfaces S_a
are precisely the collections of "honest" projective surfaces whose
Euler characteristics and log-square sums will be computed.