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

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 $$y^{n}=f(x)$$ y n = 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 $$d\le {5}$$ d ≤ 5 .

Tropical invariants for binary quintics and reduction types of Picard curves

“…In [10], this result was extended to arbitrary complete non-archimedean fields and an easier proof was given. This paper in turn was based on [11], where skeleta of general superelliptic curves are studied. In the latter, it was shown that one can recover the skeleton from tropicalizations of certain functions in the coefficients of f (x).…”

“…We refer to M 0,n as the moduli space of smooth n-marked curves of genus zero. There is a natural compactification of this space, given by replacing smooth curves by stable curves in (11). We denote the corresponding scheme by M 0,n .…”

Tropical invariants for binary quintics and reduction types of Picard curves

“…In [Hel21b], this result was extended to arbitrary complete non-archimedean fields and an easier proof was given. This paper in turn was based on [Hel22], where skeleta of general superelliptic curves are studied. In the latter, it was shown that one can recover the skeleton from tropicalizations of certain functions in the coefficients of f (x).…”

“…We now recall from [Hel22] how the reduction type of a Picard curve y 3 (x, z) = q(x, z) can be recovered from the (4, 1)-marked tree of (q, ). We refer the reader to [Hel22, Section 1.2] for the definition of the reduction type of a curve.…”

“…We are interested in obtaining the minimal skeleton of the Berkovich analytification of such a curve, which codifies the different possible semistable models for X. The results in [Hel22] show that this skeleton can be recovered from the marked tree type of the quintic f (x, z) = (x, z) • q(x, z). More precisely, this quintic has five distinct roots, giving a metric tree with five leaves, and the root of (x, z) gives the marking.…”