Skip to main content
SpringerLink
Log in

Curves on surfaces with trivial canonical bundle

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

We survey some results concerning Severi varieties and variation in moduli of curves lying on K3 surfaces or on abelian surfaces. A number of open problems is listed and some work in progress is mentioned.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aprodu, M.: Lazarsfeld–Mukai bundles and applications. In: Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday, pp 1–23. Springer, New York (2013)

  2. Arbarello, E., Bruno, A.: Rank two vector bundles on polarized Halphen surfaces and the Gauss–Wahl map for Du Val curves. arXiv:1609.09256

  3. Arbarello, E., Bruno, A., Farkas, G., Saccà, G.: Explicit Brill–Noether–Petri general curves. Comment. Math. Helv. 91, 477–491 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arbarello, E., Bruno, A., Sernesi, E.: Mukai’s program for curves on a \(K3\) surface. Alg. Geom. 1, 532–557 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arbarello, E., Bruno, A., Sernesi, E.: On two conjectures by J.Wahl. arXiv:1507.05002v2

  6. Bryan, J., Leung, N.C.: The enumerative geometry of \(K3\) surfaces and modular forms. J. Am. Math. Soc. 13, 371–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X.: Rational curves on \(K3\) surfaces. J. Alg. Geom. 8, 245–278 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Chen, X.: A simple proof that rational curves on \(K3\) are nodal. Math. Ann. 324, 71–104 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X.: Nodal curves on \(K3\) surfaces. arXiv:1611.07423

  10. Ciliberto, C., Flamini, F., Galati, C., Knutsen, A.L.: Moduli of nodal curves on K3 surfaces. Adv. Math. 309, 624–654 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciliberto, C., Harris, J., Miranda, R.: On the surjectivity of the Wahl map. Duke Math. J. 57, 829–858 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciliberto, C., Knutsen, A.L.: On \(k\)-gonal loci in Severi varieties on general \(K3\) surfaces. J. Math. Pures Appl. 101, 473–494 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ciliberto, C., Lopez, A.F., Miranda, R.: Projective degenerations of \(K3\) surfaces, Gaussian maps and Fano threefolds. Invent. Math. 114, 641–667 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciliberto, C., Lopez, A.F., Miranda, R.: Classification of varieties with canonical curve section via Gaussian maps on canonical curves. Am. J. Math. 120, 1–21 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ciliberto, C., Pareschi, G.: Pencils of minimal degree on curves on a \(K3\) surface. J. Reine Angew. Math. 460, 15–36 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Cukierman, F., Ulmer, D.: Curves of genus \(10\) on \(K3\) surfaces. Comput. Math. 89, 81–90 (1993)

    MATH  Google Scholar 

  17. Dedieu, T., Sernesi, E.: Equigeneric and equisingular families of curves on surfaces. Publ. Mat. 61, 175–212 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Farkas, G., Ortega, A.: Higher rank Brill–Noether theory on sections of \(K3\) surfaces. Internat. J. Math 23, 227–244 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Farkas, G., Popa, M.: Effective divisors on \(M_g\), curves on \(K3\) surfaces, and the slope conjecture. J. Alg. Geom. 14, 241–267 (2005)

    Article  MATH  Google Scholar 

  20. Farkas, G., Verra, S.: Moduli of theta characteristics via Nikulin surfaces. Math. Ann. 354, 465–496 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Green, M.L., Lazarsfeld, R.: Special divisors on curves on a \(K3\) surface. Invent. Math. 89, 357–370 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hulek, K., Weintraub, S.H.: The principal degenerations of abelian surfaces and their polarisations. Math. Ann. 286, 281–307 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kemeny, M.: The moduli of singular curves on \(K3\) surfaces. J. Math. Pures Appl. 104, 882–920 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Knutsen, A.L.: On two conjectures for curves on \(K3\) surfaces. Int. J. Math. 20, 1547–1560 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Knutsen, A.L., Lelli-Chiesa, M., Mongardi, G.: Severi varieties and Brill–Noether theory of curves on abelian surfaces. J. Reine Angew. Math. doi:10.1515/crelle-2016-0029

  27. Knutsen, A.L., Lelli-Chiesa, M., Mongardi, G.: Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds. arXiv:1507.06891

  28. Knutsen, A.L., Lelli-Chiesa, M., Verra, S.: Half Nikulin surfaces and moduli of Prym curves (in preparation)

  29. Lelli-Chiesa, M.: Stability of rank-\(3\) Lazarsfeld–Mukai bundles on K3 surfaces. Proc. London Math. Soc. 107, 451–479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lange, H., Sernesi, E.: Severi varieties and branch curves of abelian surfaces of type \((1,3)\). Internat. J. Math. 13, 227–244 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lazarsfeld, R.: Brill–Noether–Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. Algebraic Geometry Proc. Tokyo/Kyoto, 334–353. Lecture Notes in Mathematics, vol. 1016. Springer, Berlin (1983)

  33. Mukai, S.: Curves and symmetric spaces I. Am. J. Math. 117, 1627–1644 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mukai, S.: Curves and \(K3\) surfaces of genus eleven. Moduli of vector bundles. Lecture Notes in Pure and Applied Mathematics, vol. 179, pp. 189–197. Dekker, London (1996)

  35. Mukai, S.: Non-abelian Brill–Noether theory and Fano 3-folds. Sugaku Exposit. 14, 125–153 (2001)

    Google Scholar 

  36. Pinkham, H.: Deformations of algebraic varieties with \(\mathbb{G}_m\)-action. Astérisque 20, 1–131 (1974)

    MATH  Google Scholar 

  37. Voisin, C.: Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface. J. Eur. Math. Soc. 4, 363–404 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141, 1163–1190 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wahl, J.: The Jacobian algebra of a graded Gorenstein singularity. Duke Math. J. 55, 843–871 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wahl, J.: Gaussian maps on algebraic curves. J. Differ. Geom. 3, 77–98 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wahl, J.: On the cohomology of the square of an ideal sheaf. J. Alg. Geom. 6, 481–511 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Università di Pisa, Largo Bruno Pontecorvo 5, 56100, Pisa, Italy

    Margherita Lelli-Chiesa

Authors
  1. Margherita Lelli-Chiesa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Margherita Lelli-Chiesa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lelli-Chiesa, M. Curves on surfaces with trivial canonical bundle. Boll Unione Mat Ital 11, 93–105 (2018). https://doi.org/10.1007/s40574-017-0126-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-017-0126-0

Keywords

Search

Navigation