Skip to main content
SpringerLink
Log in

A note on the Petri loci

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

Let \({\mathcal {M}_g}\) be the coarse moduli space of complex projective nonsingular curves of genus g. We prove that when the Brill–Noether number ρ(g, r, n) is non-negative every component of the Petri locus \({P^r_{g,n} \subset \mathcal {M}_g}\) whose general member is a curve C such that \({W^{r+1}_n(C) = \emptyset}\) , has codimension one in \({\mathcal {M}_g}\) .

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. Castorena A., Teixidor i Bigas M.: Divisorial components of the Petri locus for pencils. J. Pure Appl. Algebr. 212, 1500–1508 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Farkas G.: Gaussian maps, Gieseker–Petri loci and large theta-characteristics. J. Reine Angew. Math. 581, 151–173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Farkas G.: Rational maps between moduli spaces of curves and Gieseker–Petri divisors. J. Algebr. Geom. 19, 243–284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fulton W., Lazarsfeld R.: On the connectedness of degeneracy loci and special divisors. Acta Math. 146, 271–283 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gieseker D.: Stable curves and special divisors. Invent. Math. 66, 251–275 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grothendieck, A.: Elements de Geometrie Algebrique III, 1. Inst. de Hautes Etudes Sci. Publ. Math. 11 (1961)

  7. Hartshorne, R.: Deformation Theory. Springer GTM vol. 257 (2010)

  8. Kempf G.: Schubert Methods with an Application to Algebraic Curves. Publication of Mathematisch Centrum, Amsterdam (1972)

    Google Scholar 

  9. Kleiman S., Laksov D.: On the existence of special divisors. Am. Math. J. 94, 431–436 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lelli-Chiesa, M.: The Gieseker–Petri divisor in M g for g ≤ 13. arXiv preprint n. 1012.3061v1

  11. Steffen F.: A generalized principal ideal theorem with an application to Brill–Noether theory. Invent. Math. 132, 73–89 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Teixidor M.: The divisor of curves with a vanishing theta null. Compos. Math. 66, 15–22 (1988)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo 1, 00146, Rome, Italy

    A. Bruno & E. Sernesi

Authors
  1. A. Bruno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Sernesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Sernesi.

Additional information

A. Bruno and E. Sernesi are members of GNSAGA-INDAM.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bruno, A., Sernesi, E. A note on the Petri loci. manuscripta math. 136, 439–443 (2011). https://doi.org/10.1007/s00229-011-0450-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-011-0450-0

Mathematics Subject Classification (2000)

Search

Navigation