Quotient singularities of products of two curves
[Singularités quotients des produits de deux courbes]
Annales de l'Institut Fourier, Online first, 42 p.

Nous donnons une méthode pour résoudre une singularité quotient de surface qui se présente comme le quotient d’une action produit d’un groupe fini sur deux courbes. En caractéristique nulle, la singularité est résolue au moyen d’une fraction continue (désingularisation de Hirzebruch–Jung). Nous développons la méthode dans le cas de la caractéristique strictement positive où le carré de la caractéristique ne divise pas l’ordre du groupe.

We give a method to resolve a quotient surface singularity which arises as the quotient of a product action of a finite group on two curves. In the characteristic zero case, the singularity is resolved by means of a continued fraction, which is known as the Hirzebruch–Jung desingularization. We develop the method in the positive characteristic case where the square of the characteristic does not divide the order of the group.

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DOI : https://doi.org/10.5802/aif.3434
Classification : 14J17,  14B05,  14M25,  14G17
Mots clés : singularité de surface, caractéristique positive, quotient sauvage, désingularisation, désingularisation partielle, géométrie torique
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Mitsui, Kentaro. Quotient singularities of products of two curves. Annales de l'Institut Fourier, Online first, 42 p.

[1] Artin, Michael On isolated rational singularities of surfaces, Am. J. Math., Volume 88 (1966) no. 1, pp. 129-136 | Article | MR 199191 | Zbl 0142.18602

[2] Cossart, Vincent; Jannsen, Uwe; Saito, Shuji Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes (2013) (https://arxiv.org/abs/0905.2191)

[3] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011, xxiv+841 pages

[4] Dolgachev, Igor V. The Euler characteristic of a family of algebraic varieties, Mat. Sb., N. Ser., Volume 89(131) (1972) no. 2(10), p. 297-312, 351 | Zbl 0226.14003

[5] Hirzebruch, Friedrich Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann., Volume 126 (1953), pp. 1-22 | Article | Zbl 0093.27605

[6] Ito, Hiroyuki; Schröer, Stefan Wildly ramified actions and surfaces of general type arising from Artin–Schreier curves, Geometry and Arithmetic (EMS Series of Congress Reports), European Mathematical Society, 2012, pp. 213-241 | Zbl 1317.14090

[7] Jung, Heinrich W. E. Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umgebung einer Stelle x=a, y=b, J. für Math., Volume 133 (1908), pp. 289-314 | Zbl 39.0493.01

[8] Lipman, Joseph Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math., Inst. Hautes Étud. Sci., Volume 36 (1969), pp. 195-279 | Article | Zbl 0181.48903

[9] Lipman, Joseph Desingularization of two-dimensional schemes, Ann. Math., Volume 107 (1978) no. 1, pp. 151-207 | Article | MR 491722 | Zbl 0349.14004

[10] Lorenzini, Dino Wild quotients of products of curves, Eur. J. Math., Volume 4 (2018) no. 2, pp. 525-554 | Article | MR 3799154 | Zbl 1401.14023

[11] Ogg, Andrew P. Elliptic curves and wild ramification, Am. J. Math., Volume 89 (1967) no. 1, pp. 1-21 | MR 207694 | Zbl 0147.39803

[12] Tomaru, Tadashi On Gorenstein surface singularities with fundamental genus p f 2 which satisfy some minimality conditions, Pac. J. Math., Volume 170 (1995) no. 1, pp. 271-295 | Article | MR 1359980 | Zbl 0848.14017

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