Effective cone of the blowup of the symmetric product of a curve
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- by Antonio Laface and Luca Ugaglia;
- Proc. Amer. Math. Soc. Ser. B 11 (2024), 229-242
- DOI: https://doi.org/10.1090/bproc/196
- Published electronically: June 24, 2024
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Abstract:
Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blowup of $C^{(2)}$ at a very general point has nonpolyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.References
- 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
- Ana-Maria Castravet, Antonio Laface, Jenia Tevelev, and Luca Ugaglia, Blown-up toric surfaces with non-polyhedral effective cone, J. Reine Angew. Math. 800 (2023), 1–44. MR 4609826, DOI 10.1515/crelle-2023-0022
- 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
- Steven Dale Cutkosky and Kazuhiko Kurano, Asymptotic regularity of powers of ideals of points in a weighted projective plane, Kyoto J. Math. 51 (2011), no. 1, 25–45. MR 2784746, DOI 10.1215/0023608X-2010-019
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR 704986, DOI 10.1007/BFb0101508
- Luis Fuentes García, Seshadri constants on ruled surfaces: the rational and the elliptic cases, Manuscripta Math. 119 (2006), no. 4, 483–505. MR 2223629, DOI 10.1007/s00229-006-0629-y
- José Luis González and Kalle Karu, Some non-finitely generated Cox rings, Compos. Math. 152 (2016), no. 5, 984–996. MR 3505645, DOI 10.1112/S0010437X15007745
- Javier González Anaya, José Luis González, and Kalle Karu, Constructing non-Mori dream spaces from negative curves, J. Algebra 539 (2019), 118–137. MR 3995238, DOI 10.1016/j.jalgebra.2019.08.005
- Javier González-Anaya, José Luis González, and Kalle Karu, Curves generating extremal rays in blowups of weighted projective planes, J. Lond. Math. Soc. (2) 104 (2021), no. 3, 1342–1362. MR 4332479, DOI 10.1112/jlms.12461
- Jürgen Hausen, Simon Keicher, and Antonio Laface, On blowing up the weighted projective plane, Math. Z. 290 (2018), no. 3-4, 1339–1358. MR 3856856, DOI 10.1007/s00209-018-2065-6
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- G. McGrath, Seshadri constants on irrational surfaces, 2018. Honors Thesis.
- Rick Miranda and Ulf Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), no. 4, 537–558. MR 867347, DOI 10.1007/BF01160474
- Gian Pietro Pirola, Base number theorem for abelian varieties. An infinitesimal approach, Math. Ann. 282 (1988), no. 3, 361–368. MR 967018, DOI 10.1007/BF01460039
Bibliographic Information
- Antonio Laface
- Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concep- ción, Chile
- MR Author ID: 634848
- ORCID: 0000-0001-6926-8249
- Email: alaface@udec.cl
- Luca Ugaglia
- Affiliation: Dipartimento di Matematica e Informatica, Università degli studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
- ORCID: 0000-0002-4149-6407
- Email: luca.ugaglia@unipa.it
- Received by editor(s): November 3, 2022
- Received by editor(s) in revised form: July 17, 2023
- Published electronically: June 24, 2024
- Additional Notes: Both authors have been partially supported by Proyecto FONDECYT Regular n. 1230287, and by “Piano straordinario per il miglioramento della qualità della ricerca e dei risultati della VQR 2020-2024 - Misura A” of the University of Palermo.
The second author is member of INdAM - GNSAGA - Communicated by: Rachel Pries
- © Copyright 2024 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 11 (2024), 229-242
- MSC (2020): Primary 14C20; Secondary 14J27, 14J29
- DOI: https://doi.org/10.1090/bproc/196
- MathSciNet review: 4762685