DOI QR코드

DOI QR Code

COVERS OF ALGEBRAIC VARIETIES VI. ANGLO-AMERICAN COVERS AND (1,3)-POLARIZED ABELIAN SURFACES

  • Casnati, Gianfranco (Dipartimento di Matematica Politecnico di Torino)
  • Received : 2008.04.17
  • Published : 2012.01.01

Abstract

In the present paper we describe a class of Gorenstein, finite and at morphism ${\varrho}$: $X{\rightarrow}Y$ of degree 6 of algebraic varieties, called Anglo-American covers. We prove a general Bertini theorem for them and we give some evidence that the cover ${\varrho}$: $A{\rightarrow}\mathbb{P}_k^2$ associated general (1, 3)-polarized abelian surface is Anglo-American.

Keywords

cover of degree d;Anglo-American cover

References

  1. A. Altman and S. Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics, Vol. 146 Springer-Verlag, Berlin-New York, 1970.
  2. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves. Vol. I, Springer-Verlag, New York, 1985.
  3. G. Casnati, Covers of algebraic varieties II. Covers of degree 5 and construction of surfaces, J. Algebraic Geom. 5 (1996), no. 3, 461-477.
  4. G. Casnati, The cover associated to a (1,3)-polarized bielliptic abelian surface and its branch locus, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 2, 375-392. https://doi.org/10.1017/S0013091500020319
  5. G. Casnati, Covers of algebraic varieties IV. A Bertini theorem for Scandinavian covers, Forum Math. 13 (2001), no. 1, 21-36. https://doi.org/10.1515/FORM.2001.21
  6. G. Casnati, Covers of algebraic varieties V. Examples of covers of degree 8 and 9 as catalecticant loci, J. Pure Appl. Algebra 182 (2003), no. 1, 17-32. https://doi.org/10.1016/S0022-4049(03)00019-7
  7. G. Casnati and T. Ekedahl, Covers of algebraic varieties I. A general structure theorem, covers of degree 3, 4 and Enriques surfaces, J. Algebraic Geom. 5 (1996), no. 3, 439-460.
  8. G. Casnati and R. Notari, On some Gorenstein loci in $Hilb_6(P^4_k)$, J. Algebra 308 (2007), no. 2, 493-523. https://doi.org/10.1016/j.jalgebra.2006.09.023
  9. W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991.
  10. Ph. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978.
  11. A. Grothendieck, Techniques de construction et theoremes d'existence en geometrie algebrique, IV: les schemas de Hilbert, Extraits du seminaire Bourbaki, Secretariat Mathematique, 1961.
  12. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52.Springer-Verlag, New York-Heidelberg, 1977.
  13. V. A. Iskovskikh and Yu. G. Prokhorov, Fano Varieties, Algebraic Geometry V, A. N. Parshin and I. R. Shafarevich, Encyclopedia of Mathematical Sciences, Vol. 47, 1991.
  14. A. R. Kustin and M. Miller, Structure theory for a class of grade four Gorenstein ideals, Trans. Amer. Math. Soc. 270 (1982), no. 1, 287-307. https://doi.org/10.1090/S0002-9947-1982-0642342-4
  15. H. Lange and E. Sernesi, Severi varieties and branch curves of abelian surfaces of type (1, 3), Internat. J. Math. 13 (2002), no. 3, 227-244. https://doi.org/10.1142/S0129167X02001381
  16. A. Lascoux, Syzygies des varietes determinantales, Adv. in Math. 30 (1978), no. 3, 202-237. https://doi.org/10.1016/0001-8708(78)90037-3
  17. J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics, Vol. 165, Birkhauser, 1998.
  18. R. Miranda, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), no. 5 1123-1158. https://doi.org/10.2307/2374349
  19. D. Mumford, Lectures on Curves on an Algebraic Surface, Princeton University Press, Princeton, N.J. 1966.
  20. F. O. Schreyer, Syzygies of canonical curves and special linear series, Math. Ann. 275 (1986), no. 1, 105-137. https://doi.org/10.1007/BF01458587