WEAK SIMULTANEOUS RESOLUTION 23
simultaneously blow down. Kt Kt may be defined topologically, even for t ^ 0,
using rational cycles. Kt Kt is constant.
Here is another example. Let A be a Riemann surface of genus g 3 with a
hyperelliptic Weierstrass point q. Let iV be the line bundle -qonA. Let M be the
total space of N. Then 4 , thought of as the 0-section of N, blows down in M. Now
deform A0 := ^4 in a family such that v4, is not hyperelliptic for t ^ 0. Let Kt
denote the canonical bundle on Ar Deform N0'-= N so that -(2g 2)iV, = Kt
for all £. Then blowing down At in Mt gives a Gorenstein singularity for all /. But
for t ^ 0, ht /*0.
LEMMA
5.5. Lef X'.'Y-* T be the germ of a (flat) deformation of' (V9 p) with T
the germ of a smooth I-dimensional space. Let t be a local coordinate for T. Then
either X is the trivial deformation, or else there is a smallest integer n 1 such that
for any map g: T -* U which induces Xfrom T: % - [/,
(i) g is uniquely determined and nonzero on the nonreduced subspace
{tn+l
= 0}
and
(ii) g is the zero map on the nonr educed subspace
{tn
0}.
PROOF.
Suppose that X is not the trivial deformation. Let n be the greatest
integer such that X induces the trivial deformation on {tn 0}. We need to show
that an induced map g: T - U is uniquely determined on {tn+l = 0} (see [6, p.
102]). It suffices that for 2 m n, every automorphism of X over {tm 0} lift
to an automorphism over
{tm+l
0}. Since X is the trivial deformation over
{tm
0}, an automorphism of X is a family of automorphisms of (V, p) over
{/m
= 0}. The automorphisms of X over
{t2
= 0} are T(F, 7^), vector fields on K
Integrating a vector field extends the infinitesimal automorphism to T.
For general m 2, we prove by induction that all automorphisms of (V, p)
over
{tm
0} extend to
{7n+1
= 0}. By the induction hypothesis, the family of
automorphisms of X over
{tm~]
= 0} extends to
{tn+x
= 0 ) . Composing X over
{/m
= 0} with the inverse of this extension, we may assume that X is the trivial
family of automorphisms over
{tm~l
0}. Let V be 1-dimensional and smooth
with local coordinate e. Via e =
tm,
X over
{tm
0} may be induced from A', a
family of automorphisms of (V, p) over
{e2
= 0). Then, as for m 2, the
automorphisms of A' extend to automorphisms over T. This induces an extension
of X over
{tm
= 0} to
{/"+1
= 0}.
Recall [34, Corollary 2.12, p. 190] as follows. Our notation in this paper differs
slightly from that in [34]. Let TT\ M - V be the minimal resolution of (V, /?), a
normal two-dimensional Gorenstein singularity. Let co: 911 - Q be a 1-convex
map which is the versal deformation of M. Q is smooth. Let Ta {q E Q |
/^ : = dim H\Mq, 0) = /z0} be the simultaneous blow-down subspace of Q. Ta is
reduced. Let oa: 9ILa - Ta be the restriction of « to co-1(7;). Let v % Tfl be
the simultaneous blow-down of GJ\ia. Let/: Ta - [/induce 77-f
l
from T: 9C - [/, the
reduction of the versal deformation of (V, p). Then [34, Corollary 2.12, p. 140], /
is finite and proper. See also [4 and 53] for related constructions.
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