# Multiplicity structure of the arc space of a fat point¶

Abstract: Consider a fat point on a line, that is, the scheme defined by \(x^m = 0\). The algebra of regular functions on the arc space of this scheme can be represented as the quotient algebra of \(k[x, x', x^{(2)}, . . .]\) by all the differential consequences of \(x^m = 0\). While this local algebra has infinite dimension, it admits a natural filtration by finite dimensional algebras corresponding to the truncations of arcs. We prove the generating series of the dimensions of these algebras is equal to \(\frac{m}{1 − mt}\). This result is motivated by nonreduced version of the geometric motivic Poincare ́ series, multiplicities in differential algebra, and recent connections between the arc spaces and the Rogers-Ramanujan identities. In particular, we prove a recent conjecture of Afsharijoo about the initial ideal of the ideal of the arc space of a fat point.

This repository includes the code discussed in Section 5.

Our code is written in Macaulay2 (v1.18):

Project page created: 11/17/2021.

Project contributors: Rida Ait El Manssour, Gleb Pogudin.

Corresponding author of this page: Rida Ait El Manssour, rida.manssour@mis.mpg.de

Software used: Macaulay2 (v1.18).

System setup used: MacBook Pro with macOS Monterey 12.0.1, Processor 3,3 GHz Intel Core i5, Memory 16 GB 2133 MHz LPDDR3, Graphics Intel Iris Graphics 550 1536 MB.