Francia's flip and derived categories

Flops, flips and perverse point sheaves on threefold stacks

We consider two cases of birational transformations of Q-Gorenstein threefolds: the case of a terminal flop and the case of a Francia flip. In the first case we show that, if one replaces the threefolds by their canonical covering stacks, the the flop coincides with the moduli stack of perverse point sheaves. This adds to a result of Kawamata showing the derived equivalence. Our construction has the drawback that the moduli space is defined using a presentation of the stack. In the second case we show that, if one takes a Francia flip $X \to Y$ and replaces $X$ by its canonical covering stack,…

论广义Grassmannian翻转的导出范畴

On Derived Categories of Generalized Grassmannian Flips

本文将标准翻转构造推广到任意广义Grassmannian，构造并分类广义Grassmannian翻转，并借助突变技术与国际象棋博弈方法证明了Sp(6,C)的9维情形满足DK翻转猜想。

Derived categories of Thaddeus pair moduli spaces via d-critical flips

We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence of fully-faithful functors between derived categories of coherent sheaves on these moduli spaces. Our result gives an evidence of a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also a categorification of wall-crossing formula of Donaldson-Thomas type invariants on ADHM sheaves introduced by Diaconescu.

Derived categories of flips and cubic hypersurfaces

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at leas…

Survey on homological flips and homological flops

We give a survey for the results in [Yeu20a, Yeu20b, Yeu20c], which attempts to relate the derived categories under general classes of flips and flops. We indicate how the approach fails because of what appears to be a formal problem. We give some ideas, and record some failed attempts, to fix this problem. We also present some new examples.

局部自由商层的 Quot 概型的导出范畴

Derived categories of Quot schemes of locally free quotients

本文研究层上局部自由商层 Quot 概型的导出范畴，提出结构猜想并在多种情形下验证，从而推广已知公式并发现虚拟翻转等新现象。