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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

The equation xm = 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),] by all differential consequences of xm = 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 mt). 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.

differential algebra, motivic Poincaré series, partition identities
Mathematical Subject Classification
Primary: 12H05, 13D40
Secondary: 05A17
Received: 6 August 2022
Revised: 25 April 2023
Accepted: 14 June 2023
Published: 16 April 2024
Rida Ait El Manssour
Université Paris Cité
Inselstraße 22
Gleb Pogudin
École Polytechnique
Institute Polytechnique de Paris

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