Multiplicity structure of the arc space of a fat point

Ait El Manssour, Rida; Pogudin, Gleb

doi:10.2140/ant.2024.18.947

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Multiplicity structure of the arc space of a fat point

Rida Ait El Manssour and Gleb Pogudin

Vol. 18 (2024), No. 5, 947–967

DOI: 10.2140/ant.2024.18.947

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 $k [x, x^{'}, x^{(2)}, \dots]$ 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 $m ∕ (1 - m t)$ . We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by the 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.

Keywords

differential algebra, motivic Poincaré series, partition identities

Mathematical Subject Classification

Primary: 12H05, 13D40

Secondary: 05A17

Milestones

Received: 6 August 2022

Revised: 25 April 2023

Accepted: 14 June 2023

Published: 16 April 2024

Authors

Rida Ait El Manssour
CNRS IRIF Université Paris Cité Paris France	MPI MiS Inselstraße 22 Leipzig Germany

Gleb Pogudin
LIX CNRS École Polytechnique Institute Polytechnique de Paris Palaiseau France

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Vol. 18, No. 5, 2024