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PreprintCC BY 4.0arXiv

Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant

  • Jesse Chien
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Abstract

Let X be an abelian surface of Picard rank 1 with ample generator H. We show that, for a Mukai vector v=(r,mH,χ) with r,m≥ 1 and χ≤ 0, the first Brill-Noether loci {E | H0(E)≠ 0} in MX(v) is nonempty of the expected codimension 1−χ. As an application, we study the rank 1 Segre invariant on these abelian surfaces. Specifically, for v a Mukai vector of a μ-stable sheaf of rank≥ 2, we compute the Segre invariant of the general sheaf in MX(v), and we determine the numbers which arise as the Segre invariant of a sheaf in MX(v).

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Cite

BibTeXRIS
BibTeX
@misc{chien2026codimension,
  title = {{Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant}},
  author = {Jesse Chien},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24251},
  eprint = {2610.24251},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24251},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant
AU  - Jesse Chien
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24251
UR  - https://arxiv.org/abs/2610.24251
AB  - Let X be an abelian surface of Picard rank 1 with ample generator H. We show that, for a Mukai vector v=(r,mH,χ) with r,m≥ 1 and χ≤ 0, the first Brill-Noether loci {E | H^0(E)≠ 0} in M_X(v) is nonempty of the expected codimension 1−χ. As an application, we study the rank 1 Segre invariant on these abelian surfaces. Specifically, for v a Mukai vector of a μ-stable sheaf of rank≥ 2, we compute the Segre invariant of the general sheaf in M_X(v), and we determine the numbers which arise as the Segre invariant of a sheaf in M_X(v).
ER  -