The Geometry of Rings of Components of Hurwitz Spaces
Abstract
We study a variant of the ring of components of Hurwitz moduli spaces for covers, introduced by Ellenberg, Venkatesh and Westerland in 2016 in their proof of the Cohen-Lenstra conjecture for function fields. For $G$-covers of the projective line, we show that the ring of components is a commutative algebra of finite type. We therefore study it using tools from algebraic geometry. We obtain a description of the spectrum, relating its geometry to group-theoretical properties and combinatorial aspects of Galois covers. When $G$ is a symmetric group, we are able to fully describe the geometric points of the spectrum.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.12793
- arXiv:
- arXiv:2210.12793
- Bibcode:
- 2022arXiv221012793S
- Keywords:
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- Mathematics - Number Theory;
- 14H10;
- 14H30;
- 14D22;
- 32G15;
- 12F12;
- 11G20
- E-Print:
- 52 pages, 3 figures. Comments welcome