MIXED QUASI-ÉTALE QUOTIENTS WITH ARBITRARY SINGULARITIES

FRAPPORTI, DAVIDE ; PIGNATELLI, ROBERTO

Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 143-165

Résumé

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with pg = q = 1 and Albanese fibre of genus bigger than K2.

Détail

DOI : 10.1017/S0017089514000184

Mots-clés : 14J29, 58E40, 14Q10

Bibliographie

[1] 1.Armstrong, M. A., On the fundamental group of an orbit space, Proc. Camb. Phil. Soc. 61 (1965), 639–646. Google Scholar

[2] 2.Armstrong, M. A., The fundamental group of the orbit space of a discontinuous group, Proc. Camb. Phil. Soc. 64 (1968), 299–301. Google Scholar

[3] 3.Barth, A. P., Hulek, K., Peters, C. A. M. and Van De Ven, A., Compact {c}omplex {s}urfaces, vol. 4 (Springer, {B}erlin, Germany, 2004) (Ergebnisse der Mathematik und ihrer Grenzgebiete). Google Scholar

[4] 4.Bauer, I. and Catanese, F., Some new surfaces with p = q = 0, in The Fano Conference (Alberto, C., Alberto, C. and Marina, M., Editors) (Universita di Torino, Dipartimento di Matematica, Turin, 2004), 123–142. Google Scholar

[5] 5.Bauer, I., Catanese, F. and Grunewald, F., The classification of surfaces with p = q = 0 isogenous to a product of curves, Pure Appl. Math. Q. 4 (2) (2008), 547–586. Google Scholar | DOI

[6] 6.Bauer, I., Catanese, F., Grunewald, F. and Pignatelli, R., Quotients of products of curves, new surfaces with p = 0 and their fundamental groups, Am. J. Math. 134 (4) (2012), 993–1049. Google Scholar

[7] 7.Bauer, I., Catanese, F. and Pignatelli, R., Complex {s}urfaces of {g}eneral {t}ype: {S}ome {r}ecent {p}rogress, in Global aspects of complex geometry (Springer, Berlin, Germany, 2006), pp. 1–58. Google Scholar

[8] 8.Bauer, I., Catanese, F. and Pignatelli, R., Surfaces of general type with geometric genus zero: A survey, in Complex and differential geometry, vol. 8, Springer Proceedings in Mathematics (Springer, Berlin, Germany, 2011), pp. 1–48. Google Scholar

[9] 9.Bauer, I. and Pignatelli, R., The classification of minimal product-quotient surfaces with p = 0, Math. Comput. 81 (280) (2012), 2389–2418. Google Scholar

[10] 10.Bauer, I. and Pignatelli, R., Product-quotient surfaces: New invariants and algorithms. arXiv:1308.5508 [math.AG] (preprint 2013). Google Scholar

[11] 11.Beauville, A., Surfaces algébriques complexes} (Société Mathématique de France, Paris, 1978). Google Scholar

[12] 12.Beauville, A., L' inégalité p ≥ 2q − 4 pour les surfaces de type général, Bull. Soc. Math. France 110 (3) (1982), 343–346. Google Scholar

[13] 13.Bombieri, E., Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. Google Scholar

[14] 14.Brieskorn, E., Rationale {singularitäten} komplexer {F}lächen, Invent. Math. 4 (1968), 336–358. Google Scholar

[15] 15.Carnovale, G. and Polizzi, F., The classification of surfaces with p = q = 1 isogenous to a product of curves, Adv. Geom. 9 (2) (2009), 233–256. Google Scholar

[16] 16.Catanese, F., Fibred surfaces, varieties isogenous to a product and related moduli spaces, Am. J. Math. 122 (1) (2000), 1–44. Google Scholar | DOI

[17] 17.Catanese, F., {Q.E.D.} for algebraic varieties, J. Differ. Geom. 77 (1) (2007), 43–75. Google Scholar

[18] 18.Catanese, F., On a class of surfaces of general type, in Algebraic surfaces, {C.I.M.E. Summer Sch.}, vol. 76 (Springer, Heidelberg, Germany, 2010), pp. 267–284. Google Scholar

[19] 19.Catanese, F. and Ciliberto, C., Surfaces with p = q = 1, in Problems in the theory of surfaces and their classification (Cortona, 1988) (Catanese, F.et al., Editors), Sympos. Math., XXXII (Academic Press, London, 1991), pp. 49–79. Google Scholar

[20] 20.Catanese, F. and Ciliberto, C., Symmetric products of elliptic curves and surfaces of general type with p = q = 1. J. Algebr. Geom. 2 (3) (1993), 389–411. Google Scholar

[21] 21.Catanese, F., Ciliberto, C. and Lopes, M Mendes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc. 350 (1) (1998), 275–308. Google Scholar

[22] 22.Catanese, F. and Pignatelli, R., Fibrations of low genus. {I}, Ann. Sci. École Norm. Sup. (4) 39 (6) (2006), 1011–1049. Google Scholar | DOI

[23] 23.Cossec, F. and Dolgachev, I., Enriques Surfaces {I} (Birkhäuser, Heidelberg, Germany, 1989). Google Scholar

[24] 24.Frapporti, D., Mixed surfaces, new surfaces of general type with p = 0 and their fundamental group, Collect. Math. 64 (3) (2013), 293–311. Google Scholar

[25] 25.Hacon, C. D. and Pardini, R., Surfaces with p = q = 3, Trans. Amer. Math. Soc. 354 (7), (2002), 2631–2638 (electronics). Google Scholar

[26] 26.Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput 24 (3–4) (1997), 235–265. Google Scholar | DOI

[27] 27.Matsuki, K., Introduction to the {M}ori program, {Graduate Texts in Mathematics}, vol. 56 (Springer-{V}erlag, Heidelberg, Germany, 2002). Google Scholar | DOI

[28] 28.Mistretta, E. and Polizzi, F., Standard isotrivial fibrations with p = q = 1, {II}, J. Pure Appl. Algebr. 214 (4) (2010), 344–369. Google Scholar | DOI

[29] 29.Penegini, M., The classification of isotrivially fibred surfaces with {p = q = 2, Collect. Math. 62 (3) (2011), 239–274. (With an appendix by Sönke Rollenske.) Google Scholar

[30] 30.Pignatelli, R., Some (big) irreducible components of the moduli space of minimal surfaces of general type with {p = q = 1 and K 2 = 4, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (3) (2009), 207–226. Google Scholar

[31] 31.Pirola, G. P., Surfaces with {p = q = 3, Manuscr. Math. 108 (2) (2002), 163–170. Google Scholar

[32] 32.Polizzi, F., On surfaces of general type with p = q = 1 isogenous to a product of curves, Comm. Algebra 36 (6) (2008), 2023–2053. Google Scholar

[33] 33.Polizzi, F., Standard isotrivial fibrations with p = q = 1, J. Algebra 321 (6) (2009), 1600–1631. Google Scholar

[34] 34.Polizzi, F., Numerical properties of isotrivial fibrations, Geom. Dedicata 147 (2010), 323–355. Google Scholar | DOI

[35] 35.Reid, M., Young person's guide to canonical singularities, in Algebraic geometry – {B}owdoin 1985, Part 1 (Brunswick, Maine, 1985), {Proc. Sympos. Pure Math.}, vol. 46 (American Mathematical Society, Providence, RI, 1987), pp. 345–414. Google Scholar

[36] 36.Rito, C., On surfaces with {p = q = 1 and non-ruled bicanonial involution, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (1) (2007), 81–102. Google Scholar

[37] 37.Rito, C., Involutions on surfaces with {p = q = 1, Collect. Math. 61 (1) (2010), 81–106. Google Scholar

[38] 38.Rito, C., On equations of double planes with {p = q = 1, Math. Comp. 79 (270) (2010), 1091–1108. Google Scholar | DOI

[39] 39.Serrano, F., Isotrivial fibred surfaces, Ann. Mat. Pura Appl. 171 (4) (1996), 63–81. Google Scholar

[40] 40.Wiman, A., Über die hyperelliptischen {K}urven und diejenigen vom {G}eschlechte p = 3, welche eindeutige {t}ransformationen in sich zulassen, {B}ihang {K}ongl. {S}venska {V}etenkamps-{A}kademiens {H}andlingar 21 (1895), 1–23. Google Scholar

[41] 41.Zucconi, F., Surfaces with p = q = 2 and an irrational pencil, Canad. J. Math. 55 (3) (2003), 649–672. Google Scholar | DOI