On surfaces with p g =q=1 and non-ruled bicanonical involution
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 81-102.

This paper classifies surfaces S of general type with p g =q=1 having an involution i such that S/i has non-negative Kodaira dimension and that the bicanonical map of S factors through the double cover induced by i. It is shown that S/i is regular and either: a) the Albanese fibration of S is of genus 2 or b) S has no genus 2 fibration and S/i is birational to a K3 surface. For case a) a list of possibilities and examples are given. An example for case b) with K 2 =6 is also constructed.

Classification: 14J29
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     title = {On surfaces with $p_g = q = 1$ and non-ruled bicanonical involution},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Rito, Carlos. On surfaces with $p_g = q = 1$ and non-ruled bicanonical involution. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 81-102. http://www.numdam.org/item/ASNSP_2007_5_6_1_81_0/

[1] A. Beauville, “Surfaces Algébriques Complexes”, Astérisque 54, 1978. | Numdam | Zbl

[2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. Inst. Hautes Étud. Sci. 42 (1972), 171-219. | Numdam | MR | Zbl

[3] G. Borrelli, On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, preprint, math.AG/0312351. | MR | Zbl

[4] W. Barth, C. Peters and A. Van De Ven, “Compact Complex Surfaces”, Ergebn. der Math. 3, Folge, Band 4, Springer-Verlag, Berlin, 1984. | MR | Zbl

[5] F. Catanese, On a class of surfaces of general type, In: “Algebraic Surfaces”, CIME, Liguori, 1981, 269-284.

[6] F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, Amer. Math. Soc. Contemp. Math. 241 (1999), 97-120. | MR | Zbl

[7] F. Catanese and C. Ciliberto, Surfaces with p g =q=1, In: Sympos. Math., Vol. 32, Academic Press, 1991, 49-79. | MR | Zbl

[8] F. Catanese, C. Ciliberto and M. Mendes Lopes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc. 350 (1988), 275-308. | MR | Zbl

[9] A. Calabri, C. Ciliberto and M. Mendes Lopes Numerical Godeaux surfaces with an involution, Trans. Amer. Math. Soc., to appear. | MR | Zbl

[10] C. Ciliberto, P. Francia and M. Mendes Lopes, Remarks on the bicanonical map for surfaces of general type, Math. Z. 224 (1997), 137-166. | MR | Zbl

[11] C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Sympos. Pure Math. 62 (1997), 57-84. | MR | Zbl

[12] C. Ciliberto and M. Mendes Lopes, On surfaces with p g =q=2 and non-birational bicanonical map, Adv. Geom. 2 (2002), 281-300. | MR | Zbl

[13] C. Ciliberto and M. Mendes Lopes, On surfaces with p g =2,q=1 and non-birational bicanonical map, In: “Algebraic Geometry”, Beltrametti, Mauro C. et al. (eds.), a volume in memory of Paolo Francia, de Gruyter, Berlin, 2002, 117-126. | MR | Zbl

[14] O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. Fr. 110 (1982), 319-346. | Numdam | Zbl

[15] B. Saint-Donat, Projective models of K-3 surfaces, Amer. J. Math. 96 (1974), 602-639. | MR | Zbl

[16] P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. Math. 4 (1952), 204-221. | MR | Zbl

[17] P. Griffiths and J. Harris, “Principles of Algebraic Geometry”, Wiley Classics Library, New York, 1994. | MR | Zbl

[18] K. Konno, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 575-595. | Numdam | MR | Zbl

[19] F. Polizzi, On surfaces of general type with p g =q=1,K 2 =3, Collect. Math. 56 (2005), 181-234. | MR | Zbl

[20] F. Polizzi, Surfaces of general type with p g =q=1,K 2 =8 and bicanonical map of degree 2, Trans. Amer. Math. Soc. 358 (2006), 759-798. | MR | Zbl

[21] M. Reid, Graded rings and birational geometry, In: “Proceedings of Algebraic Symposium” (Kinosaki, Oct 2000), K. Ohno (ed.) 1-72, available from www.maths. warwick.ac.uk/ miles/3folds

[22] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309-316. | MR | Zbl

[23] A. Todorov, A construction of surfaces with p g =1, q=0 and 2K 2 8. Counter examples of the global Torelli theorem, Invent. Math. 63 (1981), 287-304. | MR | Zbl

[24] G. Xiao, “Surfaces fibrées en courbes de genre deux”, Lecture Notes in Mathematics, Vol. 1137, Springer-Verlag, Berlin, 1985. | MR | Zbl

[25] G. Xiao, Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), 713-736. | MR | Zbl