Slicing up multigraded linear series

TL;DR

This paper explores the structure of multigraded linear series, focusing on cornering morphisms and their injectivity, with applications to algebraic geometry and representation theory.

Contribution

It introduces a condition for the injectivity of product morphisms in multigraded linear series and applies it to diverse geometric and algebraic examples.

Findings

Established a criterion for injectivity of cornering morphisms.

Applied the criterion to modules over reconstruction algebras.

Extended the analysis to equivariant Hilbert and Quot schemes.

Abstract

Multigraded linear series generalize the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We investigate the collection of the natural cornering morphisms into elementary bigraded linear series obtained from direct summands of the original globally generated vector bundle. Our main result is a condition on the injectivity of the product morphism. We apply our result in three examples: modules over the reconstruction algebra, equivariant Hilbert and Quot schemes of quotient stacks and Kapranov's tilting bundle over the Grassmannian.