no. 2
Jet schemes of normal toric surfaces
[Espaces de jets des surfaces toriques normales]
Mourtada, Hussein1
1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot, Paris 75013, France.
Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 2, pp. 237-266

For m∈ℕ,m≥1, we determine the irreducible components of the m-th jet scheme of a normal toric surface S. We give formulas for the number of these components and their dimensions. This permits to determine the log canonical threshold of a toric surface embedded in an affine space. When m varies, these components give rise to projective systems, to which we associate a weighted oriented graph. We prove that, among toric surfaces, the data of this graph is equivalent to the data of the analytical type of S. Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for m large enough, the set of components with index of speciality 1, is in 1-1 with the set of exceptional divisors that appear on the minimal resolution of S.

Pour m∈ℕ, m≥1, nous déterminons les composantes irréductibles des espaces de m-jets d’une surface torique normales S. Nous donnons des formules pour le nombre de ces composantes et pour leurs dimensions. Ceci permet de déterminer le seuil log-canonique de la surface S plongée dans un espace affine. Quand m varie, ces composantes donnent lieu à des systèmes projectifs, auxquels nous associons un graphe orienté et pondéré. Nous démontrons que, parmi les surfaces toriques, la donnée de ce graphe est équivalente à la donnée du type analytique de S. De plus, nous classifions ces composantes irréductibles via un invariant qu’on appelle indice de spécialité. Nous démontrons que pour m assez large, l’ensemble des composantes avec un indice de spécialité égal à 1, est en correspondance bijective avec l’ensemble des diviseurs exceptionnels qui apparaissent sur la résolution minimale des singularités de S.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.24033/bsmf.2736
Classification : 14B05, 14E18, 14M25
Keywords: Espaces de jets, surfaces toriques, résolution des singularités, problème de Nash.
Mots-clés : Jets schemes, toric surfaces, resolution of singularities, Nash problem.

Mourtada, Hussein 1

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot, Paris 75013, France. 
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     title = {Jet schemes of normal toric surfaces},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {237--266},
     year = {2017},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {145},
     number = {2},
     doi = {10.24033/bsmf.2736},
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Mourtada, Hussein. Jet schemes of normal toric surfaces. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 2, pp. 237-266. doi: 10.24033/bsmf.2736

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