Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-07T06:44:09.806Z Has data issue: false hasContentIssue false

The cover associated to a (1,3)-polarized bielliptic abelian surface and its branch locus

Published online by Cambridge University Press:  20 January 2009

Gianfranco Casnati
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy E-mail address: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Barth, W., Abelian surfaces with (1, 2)-polarization, in Algebraic Geometry, Sendai, 1985 (T. Oda, ed., vol. Advanced Studies in Pure Mathematics 10, 1987).Google Scholar
2.Barth, W., Peters, C. and Van de Ven, A., Compact complex surfaces (Springer, 1984).CrossRefGoogle Scholar
3.Beauville, A., Surfaces algebriques complexes, Astérisque 54 (1978).Google Scholar
4.Birkenhake, Ch. and Lange, H., A family of abelian surfaces and curves of genus four, Manuscripta Math. 85 (1994), 393407.CrossRefGoogle Scholar
5.Birkenhake, Ch., Lange, H. and Van Straten, D., Abelian surfaces of type (1,4), Math. Ann. 285 (1989), 625646.CrossRefGoogle Scholar
6.Brieskorn, E. and Knörrer, H., Plane algebraic curves (Birkhäuser, 1986).CrossRefGoogle Scholar
7.Casnati, G., Covers of algebraic varieties II. Covers of degree 5 and construction of surfaces, J. Algebraic Geom. 5 (1996), 461477.Google Scholar
8.Casnati, G., Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction, Trans. Amer. Math. Soc. 350 (1998), 13591378.CrossRefGoogle Scholar
9.Casnati, G. and Ekedahl, T., Covers of algebraic varieties I. A general structure theorem, covers of degree 3, 4 and Enriques surfaces, J. Algebraic Geom. 5 (1996), 439460.Google Scholar
10.Gallego, F. J. and Purnaprajna, B. P., Normal presentation on elliptic ruled surfaces, J. Algebra 186 (1996), 597625.CrossRefGoogle Scholar
11.Hartshorne, R., Algebraic geometry (Springer, 1977).CrossRefGoogle Scholar
12.Hulek, K., Abelian surfaces in products of projective spaces, in Algebraic Geometry. L'Aquila, 1988 (Sommese, A. J., Biancofiore, A. and Livorni, E. L., eds., Lecture Notes in Math. 1417, 1990).Google Scholar
13.Hulek, K. and Weintraub, S. H., Bielliptic abelian surfaces, Math. Ann. 283 (1989), 411429.CrossRefGoogle Scholar
14.Lange, H. and Birkenhake, Ch., Complex abelian varieties (Springer, 1992).CrossRefGoogle Scholar
15.Miranda, R., Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), 11231158.CrossRefGoogle Scholar
16.Okonek, C., Schneider, M. and Spindler, H., Vector bundles on complex projective spaces (Birkhäuser, 1980).Google Scholar
17.Pardini, R., Triple covers in positive characteristic, Ark. Mat. 27 (1989), 319341.CrossRefGoogle Scholar
18.Ramanan, S.Ample divisorson abelian surfaces, Proc. London Math. Soc. 51 (1985), 231245.CrossRefGoogle Scholar
19.Schreyer, F. O., Syzygies of canonical curves and special linear series, Math. Ann. 275 (1986), 105137.CrossRefGoogle Scholar
20.Serrano, F., Elliptic surfaces with an ample divisor of genus two, Pacific J. Math. 152 (1992), 187199.CrossRefGoogle Scholar
21.Tokunaga, H., Triple coverings of algebraic surfaces according to the Cardano formula, J. Math. Kyoto Univ. 31 (1991), 359375.Google Scholar
22.Tovena, F., Abelian surfaces with polarization of type (1,4), in Abelian varieties, Egloffstein 1993 (Barth, W., Hulek, K. and Lange, H., eds., Walter de Gruyter, 1995).Google Scholar