This paper classifies surfaces of general type with having an involution such that has non-negative Kodaira dimension and that the bicanonical map of factors through the double cover induced by It is shown that is regular and either: a) the Albanese fibration of is of genus 2 or b) has no genus 2 fibration and is birational to a surface. For case a) a list of possibilities and examples are given. An example for case b) with is also constructed.
@article{ASNSP_2007_5_6_1_81_0, author = {Rito, Carlos}, title = {On surfaces with $p_g = q = 1$ and non-ruled bicanonical involution}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {81--102}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {1}, year = {2007}, zbl = {1180.14040}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_81_0/} }
TY - JOUR AU - Rito, Carlos TI - On surfaces with $p_g = q = 1$ and non-ruled bicanonical involution JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 81 EP - 102 VL - 6 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2007_5_6_1_81_0/ LA - en ID - ASNSP_2007_5_6_1_81_0 ER -
%0 Journal Article %A Rito, Carlos %T On surfaces with $p_g = q = 1$ and non-ruled bicanonical involution %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 81-102 %V 6 %N 1 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2007_5_6_1_81_0/ %G en %F ASNSP_2007_5_6_1_81_0
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] “Surfaces Algébriques Complexes”, Astérisque 54, 1978. | Numdam | Zbl
,[2] Canonical models of surfaces of general type, Publ. Math. Inst. Hautes Étud. Sci. 42 (1972), 171-219. | Numdam | MR | Zbl
,[3] 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] “Compact Complex Surfaces”, Ergebn. der Math. 3, Folge, Band 4, Springer-Verlag, Berlin, 1984. | MR | Zbl
, and ,[5] On a class of surfaces of general type, In: “Algebraic Surfaces”, CIME, Liguori, 1981, 269-284.
,[6] Singular bidouble covers and the construction of interesting algebraic surfaces, Amer. Math. Soc. Contemp. Math. 241 (1999), 97-120. | MR | Zbl
,[7] Surfaces with , In: Sympos. Math., Vol. 32, Academic Press, 1991, 49-79. | MR | Zbl
and ,[8] On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc. 350 (1988), 275-308. | MR | Zbl
, and ,[9] Numerical Godeaux surfaces with an involution, Trans. Amer. Math. Soc., to appear. | MR | Zbl
, and[10] Remarks on the bicanonical map for surfaces of general type, Math. Z. 224 (1997), 137-166. | MR | Zbl
, and ,[11] The bicanonical map for surfaces of general type, Proc. Sympos. Pure Math. 62 (1997), 57-84. | MR | Zbl
,[12] On surfaces with and non-birational bicanonical map, Adv. Geom. 2 (2002), 281-300. | MR | Zbl
and ,[13] On surfaces with 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
and ,[14] Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. Fr. 110 (1982), 319-346. | Numdam | Zbl
,[15] Projective models of surfaces, Amer. J. Math. 96 (1974), 602-639. | MR | Zbl
,[16] On surfaces whose canonical system is hyperelliptic, Canadian J. Math. 4 (1952), 204-221. | MR | Zbl
,[17] “Principles of Algebraic Geometry”, Wiley Classics Library, New York, 1994. | MR | Zbl
and ,[18] 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] On surfaces of general type with , Collect. Math. 56 (2005), 181-234. | MR | Zbl
,[20] Surfaces of general type with and bicanonical map of degree , Trans. Amer. Math. Soc. 358 (2006), 759-798. | MR | Zbl
,[21] 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] Vector bundles of rank and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309-316. | MR | Zbl
,[23] A construction of surfaces with , and . Counter examples of the global Torelli theorem, Invent. Math. 63 (1981), 287-304. | MR | Zbl
,[24] “Surfaces fibrées en courbes de genre deux”, Lecture Notes in Mathematics, Vol. 1137, Springer-Verlag, Berlin, 1985. | MR | Zbl
,[25] Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), 713-736. | MR | Zbl
,