A Brill–Noether theorem for (toric) surfaces
Year: 2026
Type: article
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.
Source: Advances in Geometry
Authors: Alessio Cela, Carl Lian
Institutions: University of Cambridge, Washington University in St. Louis
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W7119862813