Hermite reciprocity and Schwarzenberger bundles

TL;DR

This paper explores Hermite reciprocity through geometric and cohomological methods involving Schwarzenberger bundles, revealing new insights into secant varieties, Ulrich modules, and their applications to algebraic geometry conjectures.

Contribution

It provides a geometric interpretation of Hermite reciprocity using Schwarzenberger bundles and links it to properties of secant varieties and Ulrich modules.

Findings

Exterior powers of Schwarzenberger bundles have supernatural cohomology.

Secant varieties of rational normal curves are shown to be normal, Cohen-Macaulay, and have rational singularities.

Hermite reciprocity is equivalent to the self-duality of a specific Ulrich module.

Abstract

Hermite reciprocity refers to a series of natural isomorphisms involving compositions of symmetric, exterior, and divided powers of the standard $SL_2$-representation. We survey several equivalent constructions of these isomorphisms, as well as their recent applications to Green's Conjecture on syzygies of canonical curves. The most geometric approach to Hermite reciprocity is based on an idea of Voisin to realize certain multilinear constructions cohomologically by working on a Hilbert scheme of points. We explain how in the case of ${\bf P}^1$ this can be reformulated in terms of cohomological properties of Schwarzenberger bundles. We then proceed to study these bundles from several perspectives: We show that their exterior powers have supernatural cohomology, arising as special cases of a construction of Eisenbud and Schreyer. We recover basic properties of secant varieties $\Sigma$ of rational normal curves (normality, Cohen-Macaulayness, rational singularities) by considering their desingularizations via Schwarzenberger bundles, and applying the Kempf-Weyman geometric technique. We show that Hermite reciprocity is equivalent to the self-duality of the unique rank one Ulrich module on the affine cone $\widehat{\Sigma}$ of some secant variety, and we explain how for a Schwarzenberger bundle of rank $k$ and degree $d\ge k$, Hermite reciprocity can be viewed as the unique (up to scaling) non-zero section of $(Sym^k\mathcal{E})(-d+k-1)$.