Abstract
AbstractIn 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 .
Highlights
Let X be a smooth, proper, irreducible curve over a complete algebraically closed nonarchimedean field K
Our goal in this paper is to give similar criteria for superelliptic curves, which are given by equations of the form yn = f (x)
We will assume that n is coprime to the residue characteristic, since we can express the skeleton in terms of the tree associated to f (x)
Summary
Let X be a smooth, proper, irreducible curve over a complete algebraically closed nonarchimedean field K. Using this theorem, we find that we can completely recover the isomorphism class of the tree of a polynomial from its tropical invariants. The conditions on the tropical invariants are given by rational half-spaces as in the case of elliptic curves They arise in this paper as equations that describe moduli of Newton polygons, see Sect. As an application of Theorem 1.1, we obtain that the tropical invariants of f (x) determine the semistable reduction type of the superelliptic curve yn = f (x). Theorem 1.2 Let Xn, f be the superelliptic curve defined by yn = f (x), where f (x) is a separable polynomial. 2.5, we give polyhedral equations for various moduli of marked tree filtrations We write these down explicitly for polynomials of degree d = 3, 4, 5.
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