Segre三次空间的同调投射对偶
Homological projective duality for the Segre cubic
摘要：
Segre三次曲面和Castelnuovo-Richmond四次曲面是$\mathbb{P}^4$中的两个射影对偶超曲面，其历史可以追溯到19世纪。我们将解释如何库兹涅佐夫的理论同调投影对偶升降机这一投影对偶之间的关系，导出范畴的一个小决议的塞格雷三次和一个小决议的Coble四倍，双重覆盖的$\mathbb{P}^4$分歧沿Castelnuovo-Richmond四次。同调投射对偶然后提供了线性截面的导出范畴的描述，我们将描述它来说明理论。塞格雷三次函数和科布尔四次函数的例子足够重要，足以表现出有趣的行为，同时也足够容易解释这个特殊和非常经典的例子中的一般机制。
英文摘要：
The Segre cubic and Castelnuovo-Richmond quartic are two projectively dual hypersurfaces in $\mathbb{P}^4$, with a long and rich history starting in the 19th century. We will explain how Kuznetsov's theory of homological projective duality lifts this projective duality to a relationship between the derived category of a small resolution of the Segre cubic and a small resolution of the Coble fourfold, the double cover of $\mathbb{P}^4$ ramified along the Castelnuovo-Richmond quartic. Homological projective duality then provides a description of the derived categories of linear sections, which we will describe to illustrate the theory. The case of the Segre cubic and Coble fourfold is non-trivial enough to exhibit interesting behavior, whilst being easy enough to explain the general machinery in this special and very classical case.
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