Multiplicity structure of the arc space of a fat point
Rida Ait El Manssour, Gleb Pogudin
TL;DR
The paper addresses the multiplicity structure of the arc space of a fat point $\mathcal{I}_m=\langle x^m\rangle$ on the line by studying the differential-arc algebra $k[x^{(\infty)}]/\mathcal{I}_m^{(\infty)}$ and its truncations. It proves sharp dimension formulas: $\dim_k(k[x^{(\le h)}]/\langle x^m\rangle^{(\infty)})=m^{h+1}$, yielding the rational generating series $D_{\mathcal{I}_m}(t)=\frac{m}{1- mt}$, and identifies the standard monomials for $\langle x^i,(x^m)^{(\infty)}\rangle$ as $\mathcal{F}_{i-1,m-i}$, which implies a lex-initial-ideal description independent of several orderings. The approach centers on embedding the arc algebra into a differential exterior algebra, enabling both lower and upper bounds that match exactly, and connects to motivic Poincaré series, differential-algebra multiplicities, and Rogers–Ramanujan-type identities, including validation of an Afsharijoo conjecture. Computational experiments extend the analysis to general fat points, revealing both cases where the same rational form persists and cases where it fails, and motivating several open questions about stabilization and generalizations. Overall, the work ties arc-space multiplicities to combinatorial monomial structures and differential-algebraic techniques, providing new proofs and insights into related partition identities and motivic phenomena.
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 Poincaré 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.