A codimension 2 component of the Gieseker-Petri locus
Author:
Margherita Lelli-Chiesa
Journal:
J. Algebraic Geom. 31 (2022), 751-771
DOI:
https://doi.org/10.1090/jag/780
Published electronically:
January 11, 2022
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Abstract |
References |
Additional Information
Abstract: We show that the Brill-Noether locus $M^3_{18,16}$ is an irreducible component of the Gieseker-Petri locus in genus $18$ having codimension $2$ in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.
References
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- Margherita Lelli-Chiesa, The Gieseker-Petri divisor in $\scr {M}_g$ for $g\leq 13$, Geom. Dedicata 158 (2012), 149–165. MR 2922709, DOI 10.1007/s10711-011-9626-8
- Margherita Lelli-Chiesa, Stability of rank-3 Lazarsfeld-Mukai bundles on $K3$ surfaces, Proc. Lond. Math. Soc. (3) 107 (2013), no. 2, 451–479. MR 3092344, DOI 10.1112/plms/pds087
- Margherita Lelli-Chiesa, Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison, Adv. Math. 268 (2015), 529–563. MR 3276604, DOI 10.1016/j.aim.2014.08.011
- Shigefumi Mori, On degrees and genera of curves on smooth quartic surfaces in $\textbf {P}^3$, Nagoya Math. J. 96 (1984), 127–132. MR 771073, DOI 10.1017/S0027763000021188
- Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, DOI 10.1007/BF01389137
- Giuseppe Pareschi, A proof of Lazarsfeld’s theorem on curves on $K3$ surfaces, J. Algebraic Geom. 4 (1995), no. 1, 195–200. MR 1299009
- B. Saint-Donat, Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI 10.2307/2373709
- Frauke Steffen, A generalized principal ideal theorem with an application to Brill-Noether theory, Invent. Math. 132 (1998), no. 1, 73–89. MR 1618632, DOI 10.1007/s002220050218
References
- Andrea Bruno and Edoardo Sernesi, A note on the Petri loci, Manuscripta Math. 136 (2011), no. 3-4, 439–443. MR 2844819, DOI 10.1007/s00229-011-0450-0
- Abel Castorena, Remarks on the Gieseker-Petri divisor in genus eight, Rend. Circ. Mat. Palermo (2) 59 (2010), no. 1, 143–150. MR 2639445, DOI 10.1007/s12215-010-0011-5
- Dawei Chen, Gavril Farkas, and Ian Morrison, Effective divisors on moduli spaces of curves and abelian varieties, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 131–169. MR 3114939
- David Eisenbud, Linear sections of determinantal varieties, Amer. J. Math. 110 (1988), no. 3, 541–575. MR 944327, DOI 10.2307/2374622
- David Eisenbud and Joe Harris, Irreducibility of some families of linear series with Brill-Noether number $-1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 33–53. MR 985853
- Gavril Farkas, Brill-Noether loci and the gonality stratification of $\mathcal {M}_g$, J. Reine Angew. Math. 539 (2001), 185–200. MR 1863860, DOI 10.1515/crll.2001.074
- Gavril Farkas, Gaussian maps, Gieseker-Petri loci and large theta-characteristics, J. Reine Angew. Math. 581 (2005), 151–173. MR 2132674, DOI 10.1515/crll.2005.2005.581.151
- Gavril Farkas, Rational maps between moduli spaces of curves and Gieseker-Petri divisors, J. Algebraic Geom. 19 (2010), no. 2, 243–284. MR 2580676, DOI 10.1090/S1056-3911-09-00510-4
- Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
- Daniel Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016. MR 3586372, DOI 10.1017/CBO9781316594193
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168, DOI 10.1017/CBO9780511711985
- Margherita Lelli-Chiesa, The Gieseker-Petri divisor in $\mathcal {M}_g$ for $g\leq 13$, Geom. Dedicata 158 (2012), 149–165. MR 2922709, DOI 10.1007/s10711-011-9626-8
- Margherita Lelli-Chiesa, Stability of rank-3 Lazarsfeld-Mukai bundles on $K3$ surfaces, Proc. Lond. Math. Soc. (3) 107 (2013), no. 2, 451–479. MR 3092344, DOI 10.1112/plms/pds087
- Margherita Lelli-Chiesa, Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison, Adv. Math. 268 (2015), 529–563. MR 3276604, DOI 10.1016/j.aim.2014.08.011
- Shigefumi Mori, On degrees and genera of curves on smooth quartic surfaces in $\mathbf {P}^3$, Nagoya Math. J. 96 (1984), 127–132. MR 771073, DOI 10.1017/S0027763000021188
- Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, DOI 10.1007/BF01389137
- Giuseppe Pareschi, A proof of Lazarsfeld’s theorem on curves on $K3$ surfaces, J. Algebraic Geom. 4 (1995), no. 1, 195–200. MR 1299009
- Bernard Saint-Donat, Projective models of $K3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI 10.2307/2373709
- Frauke Steffen, A generalized principal ideal theorem with an application to Brill-Noether theory, Invent. Math. 132 (1998), no. 1, 73–89. MR 1618632, DOI 10.1007/s002220050218
Additional Information
Margherita Lelli-Chiesa
Affiliation:
Università degli studi Roma Tre, Dipartimento di Matematica e Fisica, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
MR Author ID:
980304
Email:
margherita.lellichiesa@uniroma3.it
Received by editor(s):
June 8, 2020
Received by editor(s) in revised form:
December 23, 2020
Published electronically:
January 11, 2022
Additional Notes:
The author is a member of PRIN-2017 Project Moduli Theory and Birational Classification and of GNSAGA
Article copyright:
© Copyright 2022
University Press, Inc.
