Contractions and deformations

Abstract

Suppose that f is a projective birational morphism with at most onedimensional fibres between d-dimensional varieties X and Y , satisfying Rf * O X = O Y . Consider the locus L in Y over which f is not an isomorphism. Taking the schemetheoretic fibre C over any closed point of L, we construct algebras A fib and Acon which prorepresent the functors of commutative deformations of C, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras Acon recover L, and in general the commutative deformations of neither C nor the reduced fibre can do this. As the d = 3 special case, this proves the following contraction theorem: in a neighbourhood of the point, the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable.

Noncommutative enhancements of contractions

“…The following are our main results in this setting. In particular, this globalises the local noncommutative deformation theory as studied in [DW1,DW2,BB,K2].…”

“…(2.C) By 2.4 P| U is tilting, so Ext 1 U (P| U , P| U ) = 0. Since further Y is normal, f is projective birational, and P| U is generated by global sections, it follows from [DW2,4.3] that (2.C) is an isomorphism.…”

“…A benefit of our global construction of D, which we will make use of later, is that the contraction theorem in [DW2] can be globalised. Although [DW2, was stated in the one-dimensional fibre setting, the proof works word-for-word in the more general setup here.…”

“…Y is singular along a onedimensional locus Z, and above every point in Z is an irreducible curve. At every closed point z ∈ Z, [DW2] constructs a noncommutative deformation algebra A con,z , which induces a universal sheaf E z on the formal fibre above z. In the above example, at every closed point of Z away from the origin, A con,z is isomorphic to k [[x]], and at the origin it has the form shown above.…”

On the finiteness of the derived equivalence classes of some stable endomorphism rings

“…Construction of the Contraction Algebra. In the setting of 4.1, the contraction algebra attached to f has a very explicit construction, detailed in [DW2,3.5], but provided here for convenience.…”