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.