# Birational Geometry of Degenerations by Robert Friedman By Robert Friedman

This quantity provides new advancements within the thought
of degenerations. The unifying subject is the applica-
tion of suggestions from the birational geometry of
threefolds. one of the purposes of this research are
partial compactifications of moduli areas of algebraic
varieties and perception into the geometry of gentle
varieties.

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Then the dimension of every irreducible component of Hilb(Ys /8) at [Z] is at least dimk HomYk (h, Oz) + dims 8 - dimk Obs(Z). 32 Chapter I. 4) If dim[zJ Hilb(yk) = dimk Homyk(Iz , Oz) - dimk Obs(Z), then Hilb(Ys IS) -4 S is a local complete intersection morphism at [Z]. In particular it is fiat at [Z]. Proof. 4), the Hilbert scheme of Ys IS exists. 1) as: (mQ/(msQ, m~))· 9:! HomYk(Iz, Oz}. Therefore Q is the quotient of a local Os,s-algebra R which is smooth over S of relative dimension dirnk HOmyk(Iz, Oz).

Let TB be a Noetherian ring flat over B such that TB 0B k ~ T and the mB-adic topology on TB is separated. Let TB 0B A ~ TA. Extensions of the above ideals, modules or morphisms to TA or TB -ideals, modules or morphisms are denoted by a corresponding upper index. Let I AcTA be an ideal such that QA = TA /1 A is flat over A. 2 Lemma (Notation as above). Let d2 rD 2 -----+ dl . ) F 1 -----+ 9 I -----+ 0 be a free resolution of I. Then there is a free resolution such that Proof Let {xd be a basis of FI and gi = g(Xi)' QA is flat over A, thus I A 0 k ~ I.

1) particular If q and f are flat, then p is flat and p-l{O) = f-l(q-l(O)). In (q 0 j)*[O) = j*(q*[O]). 2) Assume that p is flat and f is proper. Let Xi be the irreducible components of X and Wi := f(X i ) with reduced structure, ai = deg(XdWi)' Then Proof. 1) is obvious. 2), we may localize at a generic point of q-l(O) n f(X), thus we may assume that UWi is local and has relative dimension zero over T. We may replace Y with a suitable scheme whose support is uWi . 3 Chow Varieties 43 dimension, thus ai = 0 for every i and f*[p-I(O)] = O.

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