A Brill–Noether theorem for (toric) surfaces
-
Alessio Cela
and
Carl Lian
Abstract
The classical Brill–Noether theorem states that a map from a general curve to a projective space deforms in a family of the expected dimension if its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth projective surface. Namely, a non-constant map deforms in a family of the expected dimension if its image has anti-canonical degree at least 4.
In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms of certain toric contractions. We raise the question of whether a Brill–Noether theorem could hold for toric varieties of higher dimensions.
Funding statement: Some work on this note took place at the NSF-funded “Workshop on Tevelev degrees and related topics” at the University of Illinois, Urbana-Champaign in October 2024; we thank Deniz Genlik and Felix Janda for their work in organizing this pleasant event. A.C. is supported by SNF grant P500PT-222363. C.L. has been supported by NSF Postdoctoral Fellowship DMS-2001976 and an AMS-Simons travel grant.
Acknowledgements
We thank Renzo Cavalieri, Gavril Farkas, François Greer, Xuanchun Lu, Navid Nabijou, Dhruv Ranganathan, Terry (Dekun) Song, Montserrat Teixidor i Bigas, and Sameera Vemulapalli for conversations related to these ideas, and the referee for improvements to the exposition.
Communicated by: R. Cavalieri
References
[1] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der mathematischen Wissenschaften. Springer 1985. MR770932 Zbl 0559.1401710.1007/978-1-4757-5323-3Search in Google Scholar
[2] A. Bäuerle, Sharp degree bounds for fake weighted projective spaces. Electron. J. Combin. 31 (2024), Paper No. 1.43, 16 pages. MR4707764 Zbl 1548.1415910.37236/11703Search in Google Scholar
[3] W. Buzyńska, Fake weighted projective spaces. Preprint 2008, arXiv:0805.1211Search in Google Scholar
[4] A. Cela, C. Lian, Complete quasimaps to Blℙs (ℙr Preprint 2025, arXiv:2505.14672Search in Google Scholar
[5] K. Christ, X. He, I. Tyomkin, On the Severi problem in arbitrary characteristic. Publ. Math. Inst. Hautes Études Sci. 137 (2023), 1–45. MR4588594 Zbl 1517.1401910.1007/s10240-022-00135-xSearch in Google Scholar
[6] K. Christ, X. He, I. Tyomkin, The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces. Preprint 2025, arXiv:2501.16238Search in Google Scholar
[7] C. Ciliberto, T. Dedieu, On the irreducibility of Severi varieties on K3 surfaces. Proc. Amer. Math. Soc. 147 (2019), 4233–4244. MR4002538 Zbl 1423.1419210.1090/proc/14559Search in Google Scholar
[8] T. Coates, S. Gonshaw, A. Kasprzyk, N. Nabijou, Mutations of fake weighted projective spaces. Electron. J. Combin. 21 (2014), Paper 4.14, 27 pages. MR3284063 Zbl 1432.5202310.37236/4288Search in Google Scholar
[9] H. Conrads, Weighted projective spaces and reflexive simplices. Manuscripta Math. 107 (2002), 215–227. MR1894741 Zbl 1013.5200910.1007/s002290100235Search in Google Scholar
[10] E. Cotterill, Geometry of curves with exceptional secant planes: linear series along the general curve. Math. Z. 267 (2011), 549–582. MR2776048 Zbl 1213.1406410.1007/s00209-009-0635-3Search in Google Scholar
[11] D. A. Cox, The functor of a smooth toric variety. Tohoku Math. J. (2) 47 (1995), 251–262. MR1329523 Zbl 0828.1403510.2748/tmj/1178225594Search in Google Scholar
[12] G. Farkas, Higher ramification and varieties of secant divisors on the generic curve. J. Lond. Math. Soc. (2) 78 (2008), 418–440. MR2439633 Zbl 1155.1402410.1112/jlms/jdn038Search in Google Scholar
[13] G. Farkas, Generalized de Jonquières divisors on generic curves. Stud. Univ. Babeş-Bolyai Math. 68 (2023), 13–27. MR4568275 Zbl 1563.1401210.24193/subbmath.2023.1.01Search in Google Scholar
[14] W. Fulton, Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton Univ. Press 1993. MR1234037 Zbl 0813.1403910.1515/9781400882526Search in Google Scholar
[15] J. Harris, On the Severi problem. Invent. Math. 84 (1986), 445–461. MR837522 Zbl 0596.1401710.1007/BF01388741Search in Google Scholar
[16] J. Harris, I. Morrison, Moduli of curves. Springer 1998. MR1631825 Zbl 0913.14005Search in Google Scholar
[17] A. M. Kasprzyk, Bounds on fake weighted projective space. Kodai Math. J. 32 (2009), 197–208. MR2549542 Zbl 1216.1404710.2996/kmj/1245982903Search in Google Scholar
[18] D. Testa, The irreducibility of the spaces of rational curves on del Pezzo surfaces. J. Algebraic Geom. 18 (2009), 37–61. MR2448278 Zbl 1165.1402410.1090/S1056-3911-08-00484-0Search in Google Scholar
[19] M. Ungureanu, Dimension theory and degenerations of de Jonquières divisors. Int. Math. Res. Not. 2021, no. 20, 15911–15958. MR4329888 Zbl 1492.1405510.1093/imrn/rnz267Search in Google Scholar
[20] R. Vakil, Counting curves on rational surfaces. Manuscripta Math. 102 (2000), 53–84. MR1771228 Zbl 0967.1403610.1007/s002291020053Search in Google Scholar
© 2026 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Kac–Moody symmetric spaces: arbitrary symmetrizable complex or almost split real type
- Compactified symplectic leaves in bundle moduli spaces
- Rigid-flexible values for embeddings of four-dimensional ellipsoids into almost-cubes
- A Brill–Noether theorem for (toric) surfaces
- Crepant transformation correspondence for toric stack bundles
- The Ehrhart polynomial of a matroid specializes to the β-invariant
Articles in the same Issue
- Frontmatter
- Kac–Moody symmetric spaces: arbitrary symmetrizable complex or almost split real type
- Compactified symplectic leaves in bundle moduli spaces
- Rigid-flexible values for embeddings of four-dimensional ellipsoids into almost-cubes
- A Brill–Noether theorem for (toric) surfaces
- Crepant transformation correspondence for toric stack bundles
- The Ehrhart polynomial of a matroid specializes to the β-invariant
