Lefschetz exceptional collections in $S_k$-equivariant categories of $(\mathbb{P}^n)^k$

TL;DR

This paper constructs Lefschetz exceptional collections in the $S_k$-equivariant derived categories of $(P^n)^k$, providing explicit examples and methods for cases where rectangular collections exist or minimal ones are needed.

Contribution

It introduces new constructions of Lefschetz exceptional collections in equivariant categories, including rectangular and minimal types for specific parameter cases.

Findings

Constructed rectangular Lefschetz collections for $k=3$ and $ ext{gcd}(n+1,k)=1$ when $n=1$.

Developed minimal Lefschetz collections for even $k$ when $n=1$, and for $n=2$, $k=3$.

Extended the understanding of exceptional collections in equivariant derived categories of product projective spaces.

Abstract

We consider the bounded derived category of $S_k$-equivariant coherent sheaves on $(\mathbb{P}^n)^k$. The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist. The main results of the paper include the construction of a rectangular Lefschetz exceptional collection in the case $k=3$ and in the case $n=1$ when $\mathrm{gcd}(n+1,k)=1$. We also construct minimal Lefschetz exceptional collection for $n=1$ and even $k$, and for $n=2$ and $k=3$.