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MIXED QUASI-ÉTALE QUOTIENTS WITH ARBITRARY SINGULARITIES

Published online by Cambridge University Press:  26 August 2014

DAVIDE FRAPPORTI  and
ROBERTO PIGNATELLI
DAVIDE FRAPPORTI
Affiliation:
University of Bayreuth, Lehrstuhl Mathematik VIII, Universitaetsstrasse 30, D-95447 Bayreuth, Germany e-mail: Davide.Frapporti@uni-bayreuth.de
ROBERTO PIGNATELLI
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38123 Trento, Italy e-mail: Roberto.Pignatelli@unitn.it
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Abstract

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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.

Keywords

14J2958E4014Q10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 143 - 165
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Armstrong, M. A., On the fundamental group of an orbit space, Proc. Camb. Phil. Soc. 61 (1965), 639–646.Google Scholar
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.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.Bauer, I. and Catanese, F., Some new surfaces with pg = 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.Bauer, I., Catanese, F. and Grunewald, F., The classification of surfaces with pg = q = 0 isogenous to a product of curves, Pure Appl. Math. Q. 4 (2) (2008), 547–586.CrossRefGoogle Scholar
6.Bauer, I., Catanese, F., Grunewald, F. and Pignatelli, R., Quotients of products of curves, new surfaces with pg = 0 and their fundamental groups, Am. J. Math. 134 (4) (2012), 993–1049.Google Scholar
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.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.Bauer, I. and Pignatelli, R., The classification of minimal product-quotient surfaces with pg = 0, Math. Comput. 81 (280) (2012), 2389–2418.Google Scholar
10.Bauer, I. and Pignatelli, R., Product-quotient surfaces: New invariants and algorithms. arXiv:1308.5508 [math.AG] (preprint 2013).Google Scholar
11.Beauville, A., Surfaces algébriques complexes} (Société Mathématique de France, Paris, 1978).Google Scholar
12.Beauville, A., L' inégalité pg ≥ 2q − 4 pour les surfaces de type général, Bull. Soc. Math. France 110 (3) (1982), 343–346.Google Scholar
13.Bombieri, E., Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219.Google Scholar
14.Brieskorn, E., Rationale {singularitäten} komplexer {F}lächen, Invent. Math. 4 (1968), 336–358.Google Scholar
15.Carnovale, G. and Polizzi, F., The classification of surfaces with pg = q = 1 isogenous to a product of curves, Adv. Geom. 9 (2) (2009), 233–256.Google Scholar
16.Catanese, F., Fibred surfaces, varieties isogenous to a product and related moduli spaces, Am. J. Math. 122 (1) (2000), 1–44.CrossRefGoogle Scholar
17.Catanese, F., {Q.E.D.} for algebraic varieties, J. Differ. Geom. 77 (1) (2007), 43–75.Google Scholar
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.Catanese, F. and Ciliberto, C., Surfaces with pg = 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.Catanese, F. and Ciliberto, C., Symmetric products of elliptic curves and surfaces of general type with pg = q = 1. J. Algebr. Geom. 2 (3) (1993), 389–411.Google Scholar
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.Catanese, F. and Pignatelli, R., Fibrations of low genus. {I}, Ann. Sci. École Norm. Sup. (4) 39 (6) (2006), 1011–1049.CrossRefGoogle Scholar
23.Cossec, F. and Dolgachev, I., Enriques Surfaces {I} (Birkhäuser, Heidelberg, Germany, 1989).Google Scholar
24.Frapporti, D., Mixed surfaces, new surfaces of general type with pg = 0 and their fundamental group, Collect. Math. 64 (3) (2013), 293–311.Google Scholar
25.Hacon, C. D. and Pardini, R., Surfaces with pg = q = 3, Trans. Amer. Math. Soc. 354 (7), (2002), 2631–2638 (electronics).Google Scholar
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.CrossRefGoogle Scholar
27.Matsuki, K., Introduction to the {M}ori program, {Graduate Texts in Mathematics}, vol. 56 (Springer-{V}erlag, Heidelberg, Germany, 2002).CrossRefGoogle Scholar
28.Mistretta, E. and Polizzi, F., Standard isotrivial fibrations with pg = q = 1, {II}, J. Pure Appl. Algebr. 214 (4) (2010), 344–369.CrossRefGoogle Scholar
29.Penegini, M., The classification of isotrivially fibred surfaces with {pg = q = 2, Collect. Math. 62 (3) (2011), 239–274. (With an appendix by Sönke Rollenske.)Google Scholar
30.Pignatelli, R., Some (big) irreducible components of the moduli space of minimal surfaces of general type with {pg = 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.Pirola, G. P., Surfaces with {pg = q = 3, Manuscr. Math. 108 (2) (2002), 163–170.Google Scholar
32.Polizzi, F., On surfaces of general type with pg = q = 1 isogenous to a product of curves, Comm. Algebra 36 (6) (2008), 2023–2053.Google Scholar
33.Polizzi, F., Standard isotrivial fibrations with pg = q = 1, J. Algebra 321 (6) (2009), 1600–1631.Google Scholar
34.Polizzi, F., Numerical properties of isotrivial fibrations, Geom. Dedicata 147 (2010), 323–355.CrossRefGoogle Scholar
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.Rito, C., On surfaces with {pg = q = 1 and non-ruled bicanonial involution, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (1) (2007), 81–102.Google Scholar
37.Rito, C., Involutions on surfaces with {pg = q = 1, Collect. Math. 61 (1) (2010), 81–106.Google Scholar
38.Rito, C., On equations of double planes with {pg = q = 1, Math. Comp. 79 (270) (2010), 1091–1108.CrossRefGoogle Scholar
39.Serrano, F., Isotrivial fibred surfaces, Ann. Mat. Pura Appl. 171 (4) (1996), 63–81.Google Scholar
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.Zucconi, F., Surfaces with pg = q = 2 and an irrational pencil, Canad. J. Math. 55 (3) (2003), 649–672.CrossRefGoogle Scholar