Contractions and deformations

TL;DR

This paper studies the relationship between certain algebraic deformations and geometric contractions in algebraic geometry, showing that noncommutative deformation functors can detect contraction loci in specific morphisms.

Contribution

It introduces algebras that prorepresent deformation functors, demonstrating that noncommutative deformations uniquely recover contraction loci in certain morphisms.

Findings

Algebras $A_{fib}$ and $A_{con}$ prorepresent deformation functors.

$A_{con}$ algebras recover the contraction locus L.

Noncommutative deformations can detect when a curve is contracted without contracting a divisor.

Abstract

Suppose that f is a projective birational morphism with at most one-dimensional fibres between d-dimensional varieties X and Y, satisfying ${\bf R}f_* \mathcal{O}_X = \mathcal{O}_Y$. Consider the locus L in Y over which f is not an isomorphism. Taking the scheme-theoretic fibre C over any closed point of L, we construct algebras $A_{fib}$ and $A_{con}$ which prorepresent the functors of commutative deformations of C, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras $A_{con}$ 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.