Arc spaces and wedge spaces for toric varieties

Reguera, Ana J.

doi:10.5802/aif.3568

Arc spaces and wedge spaces for toric varieties
[Espaces d’arcs et de coins pour les variétés toriques]

Reguera, Ana J.¹

¹ Dpto. de Álgebra Análisis Matemático Geometría y Topología Universidad de Valladolid Paseo Belén 7, 47011 Valladolid (Spain)

Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 2135-2183.

Résumé
Abstract

Soit $X$ une variété torique normale sur un corps parfait $k$ et soit $X_{\infty}$ son espace d’arcs. Soit $P$ un point stable torique de $X_{\infty}$ , i.e. défini par une valuation divisorielle torique $ν$ . Nous décrivons les composantes irréductibles de $Spec \hat{𝒪_{X_{\infty}, P}}$ et leur dimension respective. Cette description est déduite de l’existence d’une famille finie de variétés toriques régulières telles que tout coin centré en $P$ se relève à l’une d’elles. Comme première conséquence, nous obtenons que l’anneau $𝒪_{X_{\infty}, P}$ n’est ni analytiquement irréductible ni caténaire en général. Une deuxième conséquence est que, lorsque $X$ est $ℚ$ -Gorenstein, nous récupérons la log-discrépance de $ν$ à partir de l’espace d’arcs $X_{\infty}$ .

Let $X$ be a normal toric variety over a perfect field $k$ and let $X_{\infty}$ be its space of arcs. Let $P$ be a toric stable point of $X_{\infty}$ , i.e. defined by a toric divisorial valuation $ν$ . We describe the irreducible components of $Spec \hat{𝒪_{X_{\infty}, P}}$ and their respective dimensions. This description is derived from the existence of a finite family of regular toric varieties such that every wedge centered at $P$ lifts to some of them. As a first consequence, we obtain that, in general, the ring $𝒪_{X_{\infty}, P}$ is neither analytically irreducible nor catenary. A second consequence is that, when $X$ is $ℚ$ -Gorenstein, we recover the log discrepancy of $ν$ from the space of arcs $X_{\infty}$ .

Reçu le : 2020-01-04
Révisé le : 2022-03-23
Accepté le : 2022-05-12
Publié le : 2023-10-09

DOI : 10.5802/aif.3568

Classification : 13A18, 14B05, 14M25, 14J17, 32S05
Keywords: Space of arcs, divisorial valuation, toric variety.
Mot clés : Espace d’arcs, valuation divisorielle, variété torique.

Affiliations des auteurs :

Reguera, Ana J. ¹

¹ Dpto. de Álgebra Análisis Matemático Geometría y Topología Universidad de Valladolid Paseo Belén 7, 47011 Valladolid (Spain)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Reguera, Ana J. Arc spaces and wedge spaces for toric varieties. Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 2135-2183. doi : 10.5802/aif.3568. https://aif.centre-mersenne.org/articles/10.5802/aif.3568/

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