Jet Schemes and Their Applications in Singularities, Toric Resolutions and Integer Partitions

Mourtada, Hussein

doi:10.1007/978-3-031-31925-9_4

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2024

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Cited by 2 publications

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“…The formula above is not the only way to take the multiplicities into account. A related and more popular approach is via Arc Hilbert-Poincaré series [Mourtada 2023, §9]; see also [Mourtada 2014;Bruschek et al 2013].…”

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confidence: 99%

Multiplicity structure of the arc space of a fat point

Ait El Manssour,

Pogudin

2024

Alg. Number Th.

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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) , . . . ] 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 − 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. := k[x (⩽ℓ) ]/ k[x (⩽ℓ) ] ∩ I (∞) m , MSC2020: primary 12H05, 13D40; secondary 05A17.

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confidence: 99%