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Peer-reviewedCC BYInventiones mathematicae

The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

  • Karl Christ1
  • Xiang He2
  • Ilya Tyomkin3
  1. 1Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123, Turin, Italy
  2. 2Yau Mathematical Sciences Center, Shuangqing Complex Building, Tsinghua University, Haidian District, Beijing, 100084, China
  3. 3Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be’er Sheva, 84105, Israel
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Abstract

In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d d -sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d d , but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. Along the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest—a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.

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Cite

BibTeXRIS
BibTeX
@article{christ2026irreducibility,
  title = {{The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces}},
  author = {Karl Christ and Xiang He and Ilya Tyomkin},
  journal = {Inventiones mathematicae},
  year = {2026},
  month = may,
  doi = {10.1007/s00222-026-01430-8},
  url = {https://doi.org/10.1007/s00222-026-01430-8},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces
AU  - Karl Christ
AU  - Xiang He
AU  - Ilya Tyomkin
PY  - 2026
DA  - 2026/05/27
JO  - Inventiones mathematicae
DO  - 10.1007/s00222-026-01430-8
UR  - https://doi.org/10.1007/s00222-026-01430-8
AB  - In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d d -sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d d , but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. Along the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest—a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.
SN  - 0020-9910
SN  - 1432-1297
ER  -