A generalized Saito freeness criterion

TL;DR

This paper generalizes Saito's freeness criterion for divisors, extending its applicability to multiple polynomials, various varieties, and special cases like group actions and positive characteristic scenarios.

Contribution

It introduces a broad generalization of Saito's criterion, enabling analysis of divisor freeness in more complex and diverse algebraic geometric contexts.

Findings

Extended Saito's criterion to multiple polynomials and varieties

Applied the criterion to examples involving group actions and positive characteristic

Demonstrated the criterion's effectiveness in various algebraic geometric situations

Abstract

We establish generalizations of Saito's criterion for the freeness of divisors in projective spaces that apply both to sequences of several homogeneous polynomials and to divisors on other complete varieties. As an application, the new criterion is applied to several examples, including sequences whose polynomials depend on disjoint sets of variables, some sequences that are equivariant for the action of a linear group, blow-ups of divisors, and certain sequences of polynomials in positive characteristics.