Duality for asymptotic invariants of graded families

TL;DR

This paper explores a duality for sequences of natural numbers that interchanges subadditive and superadditive sequences, revealing new relationships between algebraic and geometric invariants such as regularity and Seshadri constants.

Contribution

It generalizes a sequence duality to graded filtrations and establishes a new duality linking Castelnuovo-Mumford regularity and jet separation sequences.

Findings

Duality interchanges subadditive and superadditive sequences.

Generalization to differentially closed graded filtrations.

Reveals reciprocity between Seshadri constants and asymptotic regularity.

Abstract

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants. We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals. In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.