Multiplicity structure of the arc space of a fat point

TL;DR

This paper investigates the algebraic and combinatorial structure of the arc space of a fat point defined by x^m=0, providing generating series, initial ideals, and connections to motivic series and identities.

Contribution

It introduces the generating series for the dimensions of truncated arc spaces and determines the initial ideal, advancing understanding of nonreduced schemes and their motivic properties.

Findings

Generated series for dimensions: m/(1 - mt)

Determined the lexicographic initial ideal

Proved a recent conjecture by Afsharijoo

Abstract

The equation $x^m = 0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x, x', x^{(2)}, \ldots]$ by all differential consequences of $x^m = 0$. This infinite-dimensional algebra admits a natural filtration by finite dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals $\frac{m}{1 - mt}$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by nonreduced version of the geometric motivic Poincar\'e series, multiplicities in differential algebra, and connections between arc spaces and the Rogers-Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context.