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 G J\ia. Let/: Ta - [/induce 77-f 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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