Wronskians form the inverse system of the arcs of a double point
Rida Ait El Manssour, Gleb Pogudin
TL;DR
This work identifies the Macaulay inverse system of the arc-ideal for a double point as the vector space spanned by all Wronskians of the variables and their derivatives, $\mathrm{Wr}(S)$ for finite $S\subset \mathbf{x}^{(\infty)}$. The authors develop a differential-operator framework with $D_{\xi, \bm{\alpha}}$ to characterize membership in the inverse system, and establish an elimination theory that reduces the problem to Hankel-type minors and triangular matrices. As a consequence, they derive exact dimension formulas for truncations $\dim k[\mathbf{x}^{(\le h)}]/(\mathcal{I}_n^{\mathrm{Arc}}\cap k[\mathbf{x}^{(\le h)}])=(n+1)^{h+1}$ and obtain a closed-form Poincaré-type series $\sum_{h\ge0} \dim(...) t^h = \frac{n+1}{1-(n+1)t}$. These results strengthen the connection between arc spaces, differential algebra, and combinatorial structures, and extend recent work on Poincaré-type series for arc-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 Poincaré-type series for such ideals.