Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

Helminck, Paul Alexander

Abstract

In this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by yn=f(x). We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree d≤5.

Citation

Helminck, P. A. (2022). Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves. Mathematische Zeitschrift, 301(2), 1259-1297. https://doi.org/10.1007/s00209-021-02959-5

• Journal Article Type: Article
• Acceptance Date: Nov 23, 2021
• Online Publication Date: Jan 16, 2022
• Publication Date: 2022-06
• Deposit Date: Feb 16, 2022
• Publicly Available Date: Feb 17, 2022
• Journal: Mathematische Zeitschrift
• Print ISSN: 0025-5874
• Electronic ISSN: 1432-1823
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 301
• Issue: 2
• Pages: 1259-1297
• DOI: https://doi.org/10.1007/s00209-021-02959-5
• Public URL: https://durham-repository.worktribe.com/output/1213897