Involutions on surfaces of general type with pg=0 I. The composed case
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Let S be a minimal surface of general type with pg(S) =q(S) = 0 having an involution σ over the field of complex numbers. It iswell known that if the bicanonical map φ of S is composed with σ, thenthe minimal resolution W of the quotient S/σ is rational or birational toan Enriques surface. In this paper we prove that the surface W of S withK2S = 5, 6, 7, 8 having an involution σ with which the bicanonical map φ of S is composed is rational. This result applies in part to surfaces S withK2S = 5 for which Ψ has degree 4 and is composed with an involution σ. Also we list the examples available in the literature for the given K2S andthe degree of φ.
Abstract
Let S be a minimal surface of general type with pg(S) =q(S) = 0 having an involution σ over the field of complex numbers. It iswell known that if the bicanonical map φ of S is composed with σ, thenthe minimal resolution W of the quotient S/σ is rational or birational toan Enriques surface. In this paper we prove that the surface W of S withK2S = 5, 6, 7, 8 having an involution σ with which the bicanonical map φ of S is composed is rational. This result applies in part to surfaces S withK2S = 5 for which Ψ has degree 4 and is composed with an involution σ. Also we list the examples available in the literature for the given K2S andthe degree of φ.
참고문헌 (32)
- I. Bauer, "Surfaces of general type with geometric genus zero: a survey", math.AG
- E. Bombieri, "Canonical models of surfaces of general type", Inst. Hautes ´Etudes Sci. Publ. Math, 42, pp.171~219 (1973)
- P. Burniat, "Sur les surfaces de genre P12 > 0", Ann. Mat. Pura Appl, 71(4), pp.1~24 (1966)
- A. Calabri, "Numerical Godeaux surfaces with an involution", Trans. Amer. Math. Soc, 359(4), pp.1605~1632 (2007)
- A. Calabri, "Involutions on numerical Campedelli surfaces", Tohoku Math. J, 60(1), pp.1~22 (2008)
- L. Campedelli, "Sopra alcuni piani doppi notevoli con curve di diramazione del decimo ordine", Atti Accad. Naz. Lincei, 15, pp.536~542 (1932)
- I. Dolgachev, "Rational surfaces with many nodes", Compositio Math, 132(3), pp.349~363 (2002)
- M. Inoue, "Some new surfaces of general type", Tokyo J. Math, 17(2), pp.295~319 (1994)
- J. Keum, "Some new surfaces of general type with pg = 0", preprint, ~ (1988)
- J. Keum, "Quotients of fake projective planes", Geom. Topol, 12(4), pp.2497~2515 (2008)
- J. Keum, "Fixed locus of an involution acting on a Godeaux surface", Math. Proc. Cambridge Philos. Soc, 129(2), pp.205~216 (2000)
- V. S. Kulikov, "Old examples and a new example of surfaces of general type with pg = 0", Izv. Ross. Akad. Nauk Ser. Mat, 68(5), pp.123~170 (2004)
- Y. Lee, "Involutions on a surface of general type with pg = q = 0, K2 = 7", Osaka J. Math
- M. Mendes Lopes, "The bicanonical map of surfaces with pg = 0 and K2 7", Bull. London Math. Soc, 33(3), pp.265~274 (2001)
- M. Mendes Lopes, "A connected component of the moduli space of surfaces with pg = 0", Topology, 40(5), pp.977~991 (2001)
- M. Mendes Lopes, "Enriques surfaces with eight nodes", Math. Z, 241(4), pp.673~683 (2002)
- M. Mendes Lopes, "A new family of surfaces with pg = 0 and K2 = 3", Ann. Sci. ´Ecole Norm. Sup, 37(4), pp.507~531 (2004)
- M. Mendes Lopes, "Surfaces of general type with pg = 0, K2 = 6 and non birational bicanonical map", Math. Ann, 329(3), pp.535~552 (2004)
- M. Mendes Lopes, "The degree of the bicanonical map of a surface with pg = 0", Proc. Amer. Math. Soc, 135(5), pp.1279~1282 (2007)
- M. Mendes Lopes, "Numerical Campedelli surfaces with fundamental group of order 9", J. Eur. Math. Soc, 10(2), pp.457~476 (2008)
- D. Naie, "Surfaces d’Enriques et une construction de surfaces de type g´en´eral avec pg = 0", Math. Z, 215(2), pp.269~280 (1994)
- R. Pardini, "The classification of double planes of general type with K2 = 8 and pg = 0", J. Algebra, 259(1), pp.95~118 (2003)
- C. A. M. Peters, "On certain examples of surfaces with pg = 0 due to Burniat", Nagoya Math. J, 66, pp.109~119 (1977)
- I. Reider, "Vector bundles of rank 2 and linear systems on algebraic surfaces", Ann. of Math, 127(2), pp.309~316 (1988)
- C. Rito, "Involutions on surfaces with pg = q = 1", Collect. Math, 61(1), pp.81~106 (2010)
- C. Rito, "Involutions on surfaces with pg = q = 0 and K2 = 3", Geom. Dedicata, 157, pp.319~330 (2012)
- C. Rito, "On surfaces with pg = q = 1 and non-ruled bicanonial involution", Ann. Sc. Norm. Super. Pisa Cl. Sci, 6(1), pp.81~102 (2007)
- C. Rito, "Some bidouble planes with pg = q = 0 and 4 K2 7", math.AG
- Y. Shin, "Involutions on surfaces of general type with pg = 0", 박사, pp.2012
- C. Werner, "On surfaces with pg = 0 and K2 = 5", Canad. Math. Bull, 53(4), pp.746~756 (2010)
- G. Xiao, "Finitude de l’application bicanonique des surfaces de type g´en´eral", Bull. Soc. Math. France, 113(1), pp.23~51 (1985)
- G. Xiao, "Degree of the bicanonical map of a surface of general type", Amer. J. Math, 112(5), pp.713~736 (1990)