Powers of binary forms and derived Hermite reciprocity

TL;DR

This paper characterizes the ideal defining the variety of powers of binary forms, showing it is generated in degree b+1 with a linear resolution, by extending Hermite reciprocity and related representation theory results.

Contribution

It generalizes Hermite reciprocity to complexes of SL_2-representations and determines the ideal's generators and resolution for powers of binary forms.

Findings

Ideal generated in degree b+1

Linear minimal free resolution established

Determined Castelnuovo--Mumford regularity

Abstract

For $a,b \ge 1$, Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X, showing that it is generated in degree b+1 and that it has a linear minimal free resolution. We do this by generalizing results of Abdesselam and Chipalkatti on an analogue of the Foulkes--Howe map and by establishing a derived analogue of the classical Hermite reciprocity theorem for complexes of ${\rm SL}_2$-representations. In our investigation, we are led to the ideal generated by the subrepresentation ${\rm Sym}^{ab}({\Bbb C}^2) \subset {\rm Sym}^a({\rm Sym}^b {\Bbb C}^2)$. We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.