Implicitization of tensor product surfaces via virtual projective resolutions

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 …

Estimates on volumes of homogeneous polynomial spaces

In this paper we develop the "local part" of our local/global approach to globally valued fields (GVFs). The "global part", which relies on these results, is developed in a subsequent paper.We study virtual divisors on projective varieties defined over a valued field $K$, as well as sub-valuations on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). We prove a Nullstellensatz-style duality between projective varieties equipped with virtual divisors (analogous to projective varieties over a plain field) and certain sub-valuations on polynomial rings over $K$ (analogous to …

What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

截断的短虚拟分解的显式构造

Explicit constructions of short virtual resolutions of truncations

本文为光滑射影环面簇提出截断概念，构造nef截断的显式胞腔分解，证明其与已知短分解一致，并展示非平凡同调。

On Virtually Cohen-Macaulay Simplicial Complexes

We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). We also show that all balanced simplicial complexes are virtually Cohen-Macaulay.

皮卡秩为2的光滑射影环面簇的对角线短分解

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

本文针对皮卡秩为2的光滑射影环面簇，用线丛直和构成的线性复形分解对角层，并据此证明 Berkesch-Erman-Smith 猜想的新情形、建立 Horrocks 型分裂判别准则，以及给出 Orlov 关于 Rouquier 维数猜想的新证明。