Connectedness of the moduli space of Artin-Schreier curves of fixed genus

Abstract

Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties, including combinatorial type and degree. We also talk about the moduli space, a geometric object that parameterizes all possible types of abstract or planar tropical curves subject to certain conditions. Our research focuses on the moduli spaces of planar tropical curves of genus one, arbitrary degree d and any number of marked, unbounded edges. We prove that these moduli spaces are connected.

Hurwitz trees and deformations of Artin-Schreier covers

“…In §3, we examine the degeneration of étale Z/p-torsors on a disc using the language of Kato's refined Swan conductors. They are crucial to our approach, and distinguish this paper from [Dan20]. Section 4 introduces the notion of Hurwitz tree and describes how to derive such a tree from a given Z/p-deformation ( §4.2), thus proving the forward direction of Theorem 2.20.…”

“…In [Dan20], to prove that AS g is connected when g is large, we construct some equidistant deformations that do not lie in p-fibers of the Oort-Sekiguchi-Suwa component. In this section, we realize these deformations in term of exact differential forms.…”

“…That motivates the study of equal characteristic deformations of these covers. In [Dan20], by explicitly constructing some local deformations, it was shown that AS g is connected when g is sufficiently large.…”

“…Observe that, when p = 5, the only cases that the theorem does not cover are g = 10 and g = 12 (AS g is empty otherwise). The techniques from [Dan20] are inadequate to study these moduli spaces, as well as other cases. In this paper, with the aim to improve that result, we study local deformations in more detail using the notion of Hurwitz tree.…”