On the irreducibility of Severi varieties on $K3$ surfaces
HTML articles powered by AMS MathViewer
- by C. Ciliberto and Th. Dedieu PDF
- Proc. Amer. Math. Soc. 147 (2019), 4233-4244 Request permission
Abstract:
Let $(S,L)$ be a polarized $K3$ surface of genus $p \geqslant 11$ such that $\textrm {Pic}(S)=\mathbf {Z}[L]$ and $\delta$ is a non-negative integer. We prove that if $p\geqslant 4\delta -3$, then the Severi variety of $\delta$-nodal curves in $|L|$ is irreducible.References
- E. Ballico, C. Fontanari, and L. Tasin, Singular curves on $K3$ surfaces, Sarajevo J. Math. 6(19) (2010), no. 2, 165–168. MR 2757611
- Lucia Caporaso and Joe Harris, Parameter spaces for curves on surfaces and enumeration of rational curves, Compositio Math. 113 (1998), no. 2, 155–208. MR 1639183, DOI 10.1023/A:1000401119940
- Xi Chen, Rational curves on $K3$ surfaces, J. Algebraic Geom. 8 (1999), no. 2, 245–278. MR 1675158
- X. Chen, Nodal curves on K3 surfaces, preprint, arXiv:1611.07423, 2016.
- L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178. MR 1859030, DOI 10.1090/S0002-9947-01-02810-0
- Ciro Ciliberto and Thomas Dedieu, On universal Severi varieties of low genus $K3$ surfaces, Math. Z. 271 (2012), no. 3-4, 953–960. MR 2945592, DOI 10.1007/s00209-011-0898-3
- Ciro Ciliberto and Andreas Leopold Knutsen, On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds, J. Math. Pures Appl. (9) 101 (2014), no. 4, 473–494. MR 3179751, DOI 10.1016/j.matpur.2013.06.010
- T. Dedieu and E. Sernesi, Equigeneric and equisingular families of curves on surfaces, Publ. Mat. 61 (2017), no. 1, 175–212. MR 3590119, DOI 10.5565/PUBLMAT_{6}1117_{0}7
- John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 237496, DOI 10.2307/2373541
- Tomás L. Gómez, Brill-Noether theory on singular curves and torsion-free sheaves on surfaces, Comm. Anal. Geom. 9 (2001), no. 4, 725–756. MR 1868919, DOI 10.4310/CAG.2001.v9.n4.a3
- Mark Green and Robert Lazarsfeld, Special divisors on curves on a $K3$ surface, Invent. Math. 89 (1987), no. 2, 357–370. MR 894384, DOI 10.1007/BF01389083
- Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
- Robin Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), no. 3, 375–386. MR 857224, DOI 10.1215/kjm/1250520873
- Thomas Keilen, Irreducibility of equisingular families of curves, Trans. Amer. Math. Soc. 355 (2003), no. 9, 3485–3512. MR 1990160, DOI 10.1090/S0002-9947-03-03304-X
- Michael Kemeny, The universal Severi variety of rational curves on K3 surfaces, Bull. Lond. Math. Soc. 45 (2013), no. 1, 159–174. MR 3033964, DOI 10.1112/blms/bds075
- Andreas Leopold Knutsen, On $k$th-order embeddings of $K3$ surfaces and Enriques surfaces, Manuscripta Math. 104 (2001), no. 2, 211–237. MR 1821184, DOI 10.1007/s002290170040
- A. L. Knutsen, M. Lelli-Chiesa, and G. Mongardi, Severi varieties and Brill–Noether theory of curves on abelian surfaces, J. Reine Angew., DOI 10.1515/crelle-2016-0029.
- Robert Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307. MR 852158
- Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603
Additional Information
- C. Ciliberto
- Affiliation: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
- MR Author ID: 49480
- Email: cilibert@mat.uniroma2.it
- Th. Dedieu
- Affiliation: Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
- MR Author ID: 890662
- Email: thomas.dedieu@math.univ-toulouse.fr
- Received by editor(s): October 15, 2018
- Received by editor(s) in revised form: January 23, 2019
- Published electronically: July 8, 2019
- Additional Notes: The first author was supported by the Italian MIUR Project PRIN 2010–2011 “Geometria delle varietà algebriche” and by GNSAGA of INdAM. He also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome “Tor Vergata”, CUP E83C18000100006.
The authors were members of project FOSICAV, which received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 652782. - Communicated by: Rachel Pries
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4233-4244
- MSC (2010): Primary 14H15, 14J28
- DOI: https://doi.org/10.1090/proc/14559
- MathSciNet review: 4002538