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Brill–Noether loci in codimension two

  • Nicola Tarasca (a1)

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.

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[AC96]Arbarello E. and Cornalba M., Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), 705749.
[Dia85]Diaz S., Exceptional Weierstrass points and the divisor on moduli space that they define, Mem. Amer. Math. Soc. 56 (1985).
[Edi92]Edidin D., The codimension-two homology of the moduli space of stable curves is algebraic, Duke Math. J. 67 (1992), 241272.
[Edi93]Edidin D., Brill–Noether theory in codimension-two, J. Algebraic Geom. 2 (1993), 2567.
[EH86]Eisenbud D. and Harris J., Limit linear series: basic theory, Invent. Math. 85 (1986), 337371.
[EH87]Eisenbud D. and Harris J., The Kodaira dimension of the moduli space of curves of genus inline-graphic$\geq $23, Invent. Math. 90 (1987), 359387.
[EH89]Eisenbud D. and Harris J., Irreducibility of some families of linear series with Brill–Noether number inline-graphic$- 1$, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 3353.
[Fab88]Faber C., Chow rings of moduli spaces of curves. PhD thesis, Amsterdam (1988).
[Fab89]Faber C., Some results on the codimension-two Chow group of the moduli space of stable curves, in Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Mathematics, vol. 1389 (Springer, Berlin, 1989), 6675.
[Fab90a]Faber C., Chow rings of moduli spaces of curves. I. The Chow ring of inline-graphic${ \overline{ \mathcal{M} } }_{3} $, Ann. of Math. (2) 132 (1990), 331419.
[Fab90b]Faber C., Chow rings of moduli spaces of curves. II. Some results on the Chow ring of inline-graphic${ \overline{ \mathcal{M} } }_{4} $, Ann. of Math. (2) 132 (1990), 421449.
[Fab99]Faber C., A conjectural description of the tautological ring of the moduli space of curves, in Moduli of curves and abelian varieties, Aspects of Mathematics, vol. E33 (Vieweg, Braunschweig, 1999), 109129.
[FP05]Faber C. and Pandharipande R., Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), 1349.
[Far09]Farkas G., The Fermat cubic and special Hurwitz loci in inline-graphic${ \overline{ \mathcal{M} } }_{g} $, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 831851.
[Ful69]Fulton W., Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542575.
[Ful98]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).
[Har84]Harris J., On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), 437466.
[HM82]Harris J. and Mumford D., On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 2388. With an appendix by William Fulton.
[HM98]Harris J. and Morrison I., Moduli of curves, Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998).
[Log03]Logan A., The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), 105138.
[Mar46]Maroni A., Le serie lineari speciali sulle curve trigonali, Ann. Mat. Pura Appl. (4) 25 (1946), 343354.
[MS86]Martens G. and Schreyer F.-O., Line bundles and syzygies of trigonal curves, Abh. Math. Semin. Univ. Hamb. 56 (1986), 169189.
[Mum77]Mumford D., Stability of projective varieties, Enseign. Math. (2) 23 (1977), 39110.
[Mum83]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), 271328.
[Sta00]Stankova-Frenkel Z. E., Moduli of trigonal curves, J. Algebraic Geom. 9 (2000), 607662.
[Ste98]Steffen F., A generalized principal ideal theorem with an application to Brill–Noether theory, Invent. Math. 132 (1998), 7389.
[Wah12]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), 547583.
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Compositio Mathematica
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