Wronskians form the inverse system of the arcs of a double point

TL;DR

This paper characterizes the inverse system of the arc scheme ideal of a double point using Wronskians, linking algebraic, combinatorial, and differential structures, and extends results on Poincaré series.

Contribution

It demonstrates that the inverse system is spanned by Wronskians, providing a new explicit description and extending previous results on Poincaré series for these ideals.

Findings

Inverse system spanned by Wronskians of variables and derivatives

Connection between arc scheme ideals and differential algebra

Extension of Poincaré series results for double point ideals

Abstract

The ideal of the arc scheme of a double point or, equivalently, the differential ideal generated by the ideal of a double point is a primary ideal in an infinite-dimensional polynomial ring supported at the origin. This ideal has a rich combinatorial structure connecting it to singularity theory, partition identities, representation theory, and differential algebra. Macaulay inverse system is a powerful tool for studying the structure of primary ideals which describes an ideal in terms of certain linear differential operators. In the present paper, we show that the inverse system of the ideal of the arc scheme of a double point is precisely a vector space spanned by all the Wronskians of the variables and their formal derivatives. We then apply this characterization to extend our recent result on Poincar\'e-type series for such ideals.