We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,Ξ}$ over $\mathbb{Z}_{(p)}$ and $\mathbb{F}_p$ respectively that in the mixed characteristic case can be considered as a compactification of the admissible cover stack parametrizing ramified covers of curves in characteristic $0$ of degree $p$ and in the equicharacteristic case compactify the space of separable maps between smooth curves of degree $p$. These Hurwitz spaces will carry a logarithmic structure and to emphasize that they are informative, we prove that in some first cases our Hurwitz spaces are log smooth. To achieve this, we introduce various Moduli spaces that parametrize Artin-Schreier covers and the locus of zeroes and poles of certain differential forms, show their smoothness and compute their dimension.
When considering a finite surjective degree p morphism of nodal curves over Z (p) , one notes that the special fiber behaves very differently compared to the generic fiber: While in characteristic 0 at each irreducible component of the source curve the cover is given by a monodromy representation, this is far from being true in characteristic p. Here, at some components the cover is separable but it might as well be the case that at some components the cover is inseparable so is given by the relative Frobenius.
Because of these complications, large parts of the mathematical community have excluded the case of degree p covers in characteristic p when studying Hurwitz spaces that are Moduli spaces that parametrize finite covers of curves. However, results from [BT20] provide criteria for which covers of degree p can be lifted from characteristic p to characteristic 0. After fixing the data of A = (h, g, N , Λ), we introduce LH Z (p) A as the closure of the Hurwitz space in characteristic 0 of covers of degree p of smooth curves of genus g by a curve of genus h with fixed ramification pattern Λ in a large Moduli space of all finite maps of curves over Z (p) . We furthermore give an explicit modular description of this space that keeps track of the lifting data that ensures which covers in characteristic p lift to characteristic 0. This data consists of a piecewise linear function on the tropicalization of the source curve and bivariant logarithmic differential forms satisfying some conditions. Our main result will be the following theorem: THEOREM. All the irreducible components of the special fiber of the logarithmic Hurwitz space LH Z (p) A are of dimension 3g -3 + N . For p = 2 and g = 0, LH Z (2) A is log smooth over Z (2) with logarithmic structure coming from N → Z (2) that sends 1 to 2.
Similarly, for A as above and Ξ carrying additional information about the ramification profile (Λ, Ξ), we also introduce the Hurwitz space LH Fp A,Ξ that compactifies the space of separable degree p covers between smooth curves with fixed ramification profile in characteristic p and show the following analogous result: THEOREM. For p = 2 and g = 0, the Hurwitz space LH F 2 A,Ξ is log smooth over F 2 with trivial logarithmic structure.
We expect that by extending our findings in Section 4 to curves of higher genus and separable but non-normal covers, we will be able to generalize the latter theorems to higher genus and degree as long as the degree is not divided by p 2 in future work.
The author wants to express his gratitude to his academic advisor Michael Temkin who suggested this problem as part of the author’s Ph.D. project, to Dan Abramovich for providing valuable feedback and to his Ph.D. brother Simon Stojkovic for many helpful discussions about logarithmic geometry.
This research was supported by ERC Consolidator Grant 770922 -BirNonArchGeom.
In Section 2, we will recall basic notions of logarithmic geometry with a focus on log curves and basic facts about the Hurwitz space in characteristic 0 together with the minimal logarithmic structure on this Moduli space making it log smooth over its base field. We will also study the stack RUB Ξ h,b that parametrizes piecewise linear functions on the dual graph of log curves. Finally, we will introduce exact and quasi-exact relative differential forms on curves and by using the Cartier operator describe them for the relative Frobenius more explicitly than in previous works.
Next, in Section 3 we will construct the Moduli spaces LH
and LH
A,Ξ which are the main object of interest in this paper and study some of their first properties. Deformations of objects inside a fixed stratum of LH Z (p) A and LH Fp A,Ξ can be described as a product of deformations of differential forms and separable covers. Therefore, in Section 4 we will study Moduli spaces of these objects. In Section 5, we use insights of the previous sections to prove our main theorem stating that in some first cases LH Z (p) A and LH Fp A,Ξ are log smooth. Finally, in the last section, Section 6 we will illustrate our findings in an example.
In this section, we will briefly recall basic notions of logarithmic geometry with a focus on curves and their Moduli space. Logarithmic structures were introduced in [Kat89], a broader introduction is given in the survey paper [ACG + 10]. To our knowledge, the most extensive discussion of basic notions can be found in the textbook [Ogu18].
DEFINITION 2.1. Let (X, O X ) be a scheme, M be a sheaf of monoids in the étale topology on X and α : M → O X be a morphism of sheaves of monoids. If the restriction of α to the subsheaf of invertible elements is an isomorphism onto its image, then α is called a logarithmic structure on M .
A morphism between two logarithmic structures M , N on (X, O X ) is a morphism of sheaves M → N that is compatible with the morphisms to O X . A morphism between logarithmic schemes (X, O X , M X ) and (Y , O Y , M Y ) is a morphism f betwee
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