Contractions and deformations
Contractions and deformations
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DOI：
10.1353/ajm.2019.0018
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发表时间：
2015-11
影响因子：
1.7
通讯作者：
W. Donovan;M. Wemyss
中科院分区：
文献类型：
--
作者：
W. Donovan;M. Wemyss
翻译后摘要：假设$f$是一个射影双有理态射，最多一维纤维之间的$d$维品种$X$和$Y$，满足${\bfR}f_*\mathcal{O}_X=\mathcal{O}_Y$。考虑$Y$中的轨迹$L$，在该轨迹上$f$不是同构。取L的任意闭点上的概型纤维C，构造了分别表示C的交换变形和约化纤维的非交换变形的函子的代数A_（\rm fib）和A_（\rm con）.我们的主要定理是代数${\rm A}_{\rm con}$恢复$L$，一般来说，$C$和约化纤维的交换变形都不能做到这一点。作为d=3的特殊情况，这证明了以下压缩定理：在点的邻域中，态射f压缩曲线而不压缩因子当且仅当约化纤维的非交换变形的函子是可表示的。
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.