Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T13:25:18.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Brill–Noether loci in codimension two

Part of: Curves

Published online by Cambridge University Press:  07 August 2013

Nicola Tarasca
Nicola Tarasca*
Affiliation:
Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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 us consider the locus in the moduli space of curves of genus $2k$ defined by curves with a pencil of degree $k$. Since the Brill–Noether number is equal to $- 2$, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

Keywords

moduli of curvesBrill–Noether theory

MSC classification

Secondary: 14H10: Families, moduli (algebraic) 14H51: Special divisors (gonality, Brill-Noether theory)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1535 - 1568
Copyright
© The Author(s) 2013 

References

Arbarello, E. and Cornalba, M., Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), 705–749.Google Scholar
Diaz, S., Exceptional Weierstrass points and the divisor on moduli space that they define, Mem. Amer. Math. Soc. 56 (1985).Google Scholar
Edidin, D., The codimension-two homology of the moduli space of stable curves is algebraic, Duke Math. J. 67 (1992), 241–272.CrossRefGoogle Scholar
Edidin, D., Brill–Noether theory in codimension-two, J. Algebraic Geom. 2 (1993), 25–67.Google Scholar
Eisenbud, D. and Harris, J., Limit linear series: basic theory, Invent. Math. 85 (1986), 337–371.Google Scholar
Eisenbud, D. and Harris, J., The Kodaira dimension of the moduli space of curves of genus $\geq $23, Invent. Math. 90 (1987), 359–387.CrossRefGoogle Scholar
Eisenbud, D. and Harris, J., Irreducibility of some families of linear series with Brill–Noether number $- 1$, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 33–53.Google Scholar
Faber, C., Chow rings of moduli spaces of curves. PhD thesis, Amsterdam (1988).Google Scholar
Faber, C., Some results on the codimension-two Chow group of the moduli space of stable curves, in Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Mathematics, vol. 1389 (Springer, Berlin, 1989), 66–75.Google Scholar
Faber, C., Chow rings of moduli spaces of curves. I. The Chow ring of ${ \overline{ \mathcal{M} } }_{3} $, Ann. of Math. (2) 132 (1990), 331–419.CrossRefGoogle Scholar
Faber, C., Chow rings of moduli spaces of curves. II. Some results on the Chow ring of ${ \overline{ \mathcal{M} } }_{4} $, Ann. of Math. (2) 132 (1990), 421–449.Google Scholar
Faber, C., A conjectural description of the tautological ring of the moduli space of curves, in Moduli of curves and abelian varieties, Aspects of Mathematics, vol. E33 (Vieweg, Braunschweig, 1999), 109–129.Google Scholar
Faber, C. and Pandharipande, R., Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), 13–49.CrossRefGoogle Scholar
Farkas, G., The Fermat cubic and special Hurwitz loci in ${ \overline{ \mathcal{M} } }_{g} $, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 831–851.Google Scholar
Fulton, W., Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542–575.CrossRefGoogle Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, second edition (Springer, Berlin, 1998).Google Scholar
Harris, J., On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), 437–466.Google Scholar
Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–88. With an appendix by William Fulton.Google Scholar
Harris, J. and Morrison, I., Moduli of curves, Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998).Google Scholar
Logan, A., The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), 105–138.Google Scholar
Maroni, A., Le serie lineari speciali sulle curve trigonali, Ann. Mat. Pura Appl. (4) 25 (1946), 343–354.Google Scholar
Martens, G. and Schreyer, F.-O., Line bundles and syzygies of trigonal curves, Abh. Math. Semin. Univ. Hamb. 56 (1986), 169–189.Google Scholar
Mumford, D., Stability of projective varieties, Enseign. Math. (2) 23 (1977), 39–110.Google Scholar
Mumford, D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, Progress in Mathematics, vol. 36 (Birkhäuser Boston, Boston, MA, 1983), 271–328.Google Scholar
Stankova-Frenkel, Z. E., Moduli of trigonal curves, J. Algebraic Geom. 9 (2000), 607–662.Google Scholar
Steffen, F., A generalized principal ideal theorem with an application to Brill–Noether theory, Invent. Math. 132 (1998), 73–89.CrossRefGoogle Scholar
Wahl, N., Homological stability for mapping class groups of surfaces, in Handbook of moduli, Vol. III, Advanced Lectures in Mathematics, vol. 26 (International Press, Boston, 2012), 547–583.Google Scholar