截断的短虚拟分解的显式构造

刻画乘积射影空间上的多分次正则性和虚拟分解

Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

通过截断的极小分次自由分解与Betti数，给出了乘积射影空间上模的多分次Castelnuovo-Mumford正则区域的新刻画与上界，并应用于完全交。

$\mathbb{P}^1\times\mathbb{P}^1$ 中点的 Hilbert-Burch 虚拟分解

Hilbert-Burch virtual resolutions for points in $\mathbb{P}^1\times\mathbb{P}^1$

本文证明了 $\mathbb{P}^1\times\mathbb{P}^1$ 中有限点集多重度正则性极小元素的虚拟分解大多属 Hilbert-Burch 类型，给出仅依赖点数的明确描述，并指出其并非总是短分解，提供了长度为 3 的充分条件。

Virtual Resolutions for a Product of Projective Spaces

Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.

一种计数到格拉斯曼流形中超曲面的映射的简便方法

A short way of counting maps to hypersurfaces in Grassmannians

该研究利用Quot概形紧化计算从亏格g光滑射影曲线到格拉斯曼流形超曲面的d次映射虚计数，并探究大次数d下该虚计数在合适超曲面条件下是否具有枚举性。

Hanlon-Hicks-Lazarev定理的一个简短证明

A short proof of the Hanlon-Hicks-Lazarev Theorem

该研究给出环面簇领域Hanlon-Hicks-Lazarev定理的简短新证明，借此证明Berkesch-Erman-Smith关于虚拟解消的猜想，还得到单纯情形对角的解消。

Homological and combinatorial aspects of virtually Cohen--Macaulay sheaves

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to …