PreprintCC BY 4.0arXiv
Brill-Noether existence with secants
Abstract
We completely characterize when a general curve of genus g admits a non-degenerate degree d map to projective space ℙr, that is k-secant along a linear space ℙs ⊂ ℙr and deforms in a smooth family of expected dimension. This gives an optimal improvement of a theorem of Farkas, who gave necessary conditions, and proved sufficiency under additional numerical assumptions. Our main theorem shows that the natural necessary conditions are in fact sufficient, yielding a generalization of the classical Brill-Noether existence theorem. The proof relies on deformation theory of stable maps, and is valid in arbitrary characteristic.
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BibTeX
@misc{cela2026brillnoether,
title = {{Brill-Noether existence with secants}},
author = {Alessio Cela and Carl Lian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26713},
eprint = {2609.26713},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26713},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Brill-Noether existence with secants
AU - Alessio Cela
AU - Carl Lian
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26713
UR - https://arxiv.org/abs/2609.26713
AB - We completely characterize when a general curve of genus g admits a non-degenerate degree d map to projective space ℙ^r, that is k-secant along a linear space ℙ^s ⊂ ℙ^r and deforms in a smooth family of expected dimension. This gives an optimal improvement of a theorem of Farkas, who gave necessary conditions, and proved sufficiency under additional numerical assumptions. Our main theorem shows that the natural necessary conditions are in fact sufficient, yielding a generalization of the classical Brill-Noether existence theorem. The proof relies on deformation theory of stable maps, and is valid in arbitrary characteristic.
ER -