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Abstract

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 ${\rm A}_{\rm fib}$ and ${\rm A}_{\rm con}$ which prorepresent the functors of commutative deformations of $C$, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras ${\rm A}_{\rm con}$ 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.

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