Bounding invariants of fat points using a coding theory construction

TL;DR

This paper explores how the minimum distance of a linear code derived from a fat points scheme in projective space constrains the shifts in its defining ideal's minimal free resolution, providing bounds especially for complete intersections.

Contribution

It establishes a new connection between coding theory and algebraic geometry, offering bounds on free resolution shifts based on code minimum distance, with improvements for complete intersections.

Findings

Minimum distance bounds the shifts in free resolutions.

Improved lower bounds for reduced complete intersection schemes.

Constraints on algebraic invariants from coding parameters.

Abstract

Let $Z \subseteq \proj{n}$ be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of $I_Z$, the defining ideal of $Z$. We investigate this relation in the case that the support of $Z$ is a complete intersection; when $Z$ is reduced and a complete intersection we give lower bounds for $d(Z)$ that improve upon known bounds.