Multiplicity structure of the arc space of a fat point
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911840902053888 |
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| author | Manssour, Rida Ait El Pogudin, Gleb |
| author_facet | Manssour, Rida Ait El Pogudin, Gleb |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10446 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Multiplicity structure of the arc space of a fat point Manssour, Rida Ait El Pogudin, Gleb Algebraic Geometry Symbolic Computation Commutative Algebra Combinatorics 12H05, 13D40, 05A17 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. |
| title | Multiplicity structure of the arc space of a fat point |
| topic | Algebraic Geometry Symbolic Computation Commutative Algebra Combinatorics 12H05, 13D40, 05A17 |
| url | https://arxiv.org/abs/2111.10446 |