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Available in print format 		Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(e) ISSN 0025-5718(p)

     

On equations of double planes with $ p_g=q=1$

Author(s): Carlos Rito.
Journal: Math. Comp. 79 (2010), 1091-1108.
MSC (2000): Primary 14J29, 14Q05
Posted: September 2, 2009
MathSciNet review: 2600557
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with $ p_g=q=1$ and $ K^2=2,\ldots,8.$ A double plane with $ K^2=8$ having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma.

References:

[BCP97]
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language., J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. MR 1484478

[Bea78]
A. Beauville, Surfaces algébriques complexes, Astérisque, vol. 54, 1978. MR 0485887 (58:5686)

[Bor07]
G. Borrelli, The classification of surfaces of general type with nonbirational bicanonical map, J. Algebraic Geom. 16 (2007), no. 4, 625-669. MR 2357686 (2008m:14074)

[BPV84]
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, vol. 4, Springer-Verlag, Berlin, 1984. MR 749574 (86c:32026)

[Cat81]
F. Catanese, On a class of surfaces of general type, Algebraic Surfaces, CIME, Liguori, 269-284, 1981.

[Cat99]
-, Singular bidouble covers and the construction of interesting algebraic surfaces, Contemp. Math. 241, Amer. Math. Soc., 97-120, 1999. MR 1718139 (2000j:14061)

[CC91]
F. Catanese and C. Ciliberto, Surfaces with $ p_g=q=1$, Sympos. Math. 32, Academic Press, 49-79, 1991. MR 1273372 (95d:14030)

[CC93]
-, Symmetric products of elliptic curves and surfaces of general type with $ p_g=q=1$, J. Algebraic Geom. 2 (1993), no. 3, 389-411. MR 1211993 (94i:14040)

[Cil97]
C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Sympos. Pure Math. 62.1, Kollár, János et al. (eds.), Algebraic geometry, 57-84, 1997. MR 1492518 (98m:14040)

[CM02]
C. Ciliberto and M. Mendes Lopes, On surfaces with $ p_g=q=2$ and non-birational bicanonical map, Adv. Geom. 2 (2002), no. 3, 281-300. MR 1924760 (2004d:14053)

[CP06]
F. Catanese and R. Pignatelli, Fibrations of low genus. I, Ann. Scient. École. Norm. Sup. 39 (2006), no. 6, 1011-1049. MR 2316980 (2008g:14014)

[Du 52]
P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. of Math. 4 (1952), 204-221. MR 0048090 (13:977c)

[Pig]
R. Pignatelli, Some (big) irreducible components of the moduli space of minimal surfaces of general type with $ p_g=q=1$ and $ k^2=4$, math. AG/0801.1112v1.

[Pol05]
F. Polizzi, On surfaces of general type with $ p_g=q=1, k^2=3$, Collect. Math. 56 (2005), no. 2, 181-234. MR 2154303 (2006d:14038)

[Pol06]
-, Surfaces of general type with $ p_g=q=1, k^2=8$ and bicanonical map of degree $ 2$, Trans. Amer. Math. Soc. 358 (2006), no. 2, 759-798. MR 2177040 (2006j:14051)

[Pol08]
-, On surfaces of general type with $ p_g=q=1$ isogenous to a product of curves, Commun. Algebra 36 (2008), no. 6, 2023-2053. MR 2418374 (2009c:14076)

[Pol09]
-, Standard isotrivial fibrations with $ p_g=q=1$, J. Algebra 321 (2009), no. 6, 1600-1631. MR 2498259

[Rei91]
M. Reid, Campedelli versus Godeaux, Sympos. Math. 32, Academic Press, 309-365, 1991. MR 1273384 (95h:14031)

[Rit07]
C. Rito, On surfaces with $ p_g=q=1$ and non-ruled bicanonical involution, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (2007), no. 1, 81-102. MR 2341516 (2008i:14058)

[Xia85]
G. Xiao, Surfaces fibrées en courbes de genre deux, vol. 1137, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. MR 872271 (88a:14042)

[Xia90]
-, Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), no. 5, 713-736. MR 1073006 (91i:14030)

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Additional Information:

Carlos Rito
Affiliation: Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, 5000-911 Vila Real, Portugal
Email: crito@utad.pt

DOI: 10.1090/S0025-5718-09-02283-2
PII: S 0025-5718(09)02283-2
Received by editor(s): April 14, 2008
Received by editor(s) in revised form: March 26, 2009
Posted: September 2, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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