Arc spaces and wedge spaces for toric varieties
Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 2135-2183.

Let X be a normal toric variety over a perfect field k and let X be its space of arcs. Let P be a toric stable point of X , i.e. defined by a toric divisorial valuation ν. We describe the irreducible components of Spec𝒪 X ,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 ,P is neither analytically irreducible nor catenary. A second consequence is that, when X is -Gorenstein, we recover the log discrepancy of ν from the space of arcs X .

Soit X une variété torique normale sur un corps parfait k et soit X son espace d’arcs. Soit P un point stable torique de X , i.e. défini par une valuation divisorielle torique ν. Nous décrivons les composantes irréductibles de Spec𝒪 X ,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 ,P n’est ni analytiquement irréductible ni caténaire en général. Une deuxième conséquence est que, lorsque X est -Gorenstein, nous récupérons la log-discrépance de ν à partir de l’espace d’arcs X .

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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.

Reguera, Ana J. 1

1 Dpto. de Álgebra Análisis Matemático Geometría y Topología Universidad de Valladolid Paseo Belén 7, 47011 Valladolid (Spain)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Reguera, Ana J. Arc spaces and wedge spaces for toric varieties. Annales de l'Institut Fourier, Volume 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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