Orlov's functors in Macaulay2

TL;DR

This paper presents algorithms for computing Orlov's functors, which embed graded singularity categories into derived categories of projective varieties, implemented in Macaulay2 for computational algebraic geometry.

Contribution

It provides the first practical algorithms for computing Orlov's functors within Macaulay2, bridging theoretical concepts with computational tools.

Findings

Algorithms successfully compute Orlov's functors in examples

Implementation in Macaulay2 enables practical applications

Enhances computational methods in algebraic geometry

Abstract

Given a commutative and graded Gorenstein ring $R$ with associated projective variety $X$, a theorem of Orlov gives fully faithful embeddings from the graded singularity category of $R$ to the derived category of $X$, or vice versa, depending on the degree of the canonical bundle of $X$. We describe algorithms for computing these embeddings that can be implemented in Macaulay2.