On symmetric products of curves
HTML articles powered by AMS MathViewer
- by F. Bastianelli PDF
- Trans. Amer. Math. Soc. 364 (2012), 2493-2519 Request permission
Abstract:
Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(2)}$ be its second symmetric product. This paper concerns the study of some attempts at extending to $C^{(2)}$ the notion of gonality. In particular, we prove that the degree of irrationality of $C^{(2)}$ is at least $g-1$ when $C$ is generic and that the minimum gonality of curves through the generic point of $C^{(2)}$ equals the gonality of $C$. In order to produce the main results we deal with correspondences on the $k$-fold symmetric product of $C$, with some interesting linear subspaces of $\mathbb {P}^n$ enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of $C^{(2)}$ when $C$ is a generic curve of genus ${6\leq g\leq 8}$.References
- A. Alzati and G. P. Pirola, On the holomorphic length of a complex projective variety, Arch. Math. (Basel) 59 (1992), no. 4, 398–402. MR 1179468, DOI 10.1007/BF01197058
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- F. Bastianelli, Remarks on the nef cone on symmetric products of curves, Manuscripta Math. 130 (2009), no. 1, 113–120. MR 2533770, DOI 10.1007/s00229-009-0274-3
- Ciro Ciliberto and Alexis Kouvidakis, On the symmetric product of a curve with general moduli, Geom. Dedicata 78 (1999), no. 3, 327–343. MR 1725369, DOI 10.1023/A:1005280023724
- C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI 10.2307/1970801
- R. Cortini, Degree of irrationality of surfaces in $\mathbb {P}^3$, Ph.D. thesis (Università degli Studi di Genova, 2000).
- Lawrence Ein and Robert Lazarsfeld, Seshadri constants on smooth surfaces, Astérisque 218 (1993), 177–186. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265313
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Phillip Griffiths and Joseph Harris, Residues and zero-cycles on algebraic varieties, Ann. of Math. (2) 108 (1978), no. 3, 461–505. MR 512429, DOI 10.2307/1971184
- Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
- V. A. Iskovskih and Ju. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86(128) (1971), 140–166 (Russian). MR 0291172
- Andreas Leopold Knutsen, Wioletta Syzdek, and Tomasz Szemberg, Moving curves and Seshadri constants, Math. Res. Lett. 16 (2009), no. 4, 711–719. MR 2525035, DOI 10.4310/MRL.2009.v16.n4.a12
- Alexis Kouvidakis, Divisors on symmetric products of curves, Trans. Amer. Math. Soc. 337 (1993), no. 1, 117–128. MR 1149124, DOI 10.1090/S0002-9947-1993-1149124-5
- Angelo Felice Lopez and Gian Pietro Pirola, On the curves through a general point of a smooth surface in $\mathbf P^3$, Math. Z. 219 (1995), no. 1, 93–106. MR 1340851, DOI 10.1007/BF02572352
- I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319–343. MR 151460, DOI 10.1016/0040-9383(62)90019-8
- Rick Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR 1326604, DOI 10.1090/gsm/005
- T. T. Moh and W. Heinzer, On the Lüroth semigroup and Weierstrass canonical divisors, J. Algebra 77 (1982), no. 1, 62–73. MR 665164, DOI 10.1016/0021-8693(82)90277-0
- D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204. MR 249428, DOI 10.1215/kjm/1250523940
- J. Ross, Seshadri constants on symmetric products of curves, Math. Res. Lett. 14 (2007), no. 1, 63–75. MR 2289620, DOI 10.4310/MRL.2007.v14.n1.a5
- Hiro-o Tokunaga and Hisao Yoshihara, Degree of irrationality of abelian surfaces, J. Algebra 174 (1995), no. 3, 1111–1121. MR 1337188, DOI 10.1006/jabr.1995.1170
- Hisao Yoshihara, Degree of irrationality of an algebraic surface, J. Algebra 167 (1994), no. 3, 634–640. MR 1287064, DOI 10.1006/jabr.1994.1206
- Hisao Yoshihara, A note on the inequality of degrees of irrationalities of algebraic surfaces, J. Algebra 207 (1998), no. 1, 272–275. MR 1643098, DOI 10.1006/jabr.1998.7464
Additional Information
- F. Bastianelli
- Affiliation: Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy
- Address at time of publication: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy
- MR Author ID: 878934
- Email: francesco.bastianelli@unipv.it, francesco.bastianelli@unimib.it
- Received by editor(s): February 2, 2010
- Received by editor(s) in revised form: April 30, 2010
- Published electronically: January 19, 2012
- Additional Notes: This work was partially supported by PRIN 2007 “Spazi di moduli e teorie di Lie”, INdAM (GNSAGA), and FAR 2008 (PV) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici”.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2493-2519
- MSC (2010): Primary 14E05, 14Q10; Secondary 14J29, 14H51, 14N05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05378-5
- MathSciNet review: 2888217