The Hurwitz existence problem in prime degree
Abstract
Let p be a prime. We prove that every compatible branch datum of degree p over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic p. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of Sp containing a common regular subgroup of order p. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree p and the prescribed three ramification profiles. The fusion and realization results of Edmonds--Kulkarni--Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann--Hurwitz locus, every corresponding connected relative Gromov--Witten invariant of P1 is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most p is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.
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BibTeX
@misc{song2026hurwitz,
title = {{The Hurwitz existence problem in prime degree}},
author = {Jijian Song and Hailin Wen and Zebao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20572},
eprint = {2609.20572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20572},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hurwitz existence problem in prime degree
AU - Jijian Song
AU - Hailin Wen
AU - Zebao Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20572
UR - https://arxiv.org/abs/2609.20572
AB - Let p be a prime. We prove that every compatible branch datum of degree p over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic p. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of S_p containing a common regular subgroup of order p. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree p and the prescribed three ramification profiles. The fusion and realization results of Edmonds--Kulkarni--Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann--Hurwitz locus, every corresponding connected relative Gromov--Witten invariant of P^1 is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most p is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.
ER -