arXiv subjects

Matthias Hippold

Publications and source records attributed to Matthias Hippold.

3 recordsLinked to original sources

arXiv2026

The Moduli Space of Twisted Exact Differential Forms on Curves in Positive Characteristic

After fixing a pattern $\mathbf{m}$ of zeroes and poles, we introduce a Moduli stack $\Gamma\mathcal{M}_{g,n}^{ex, \mathbf{m}}$ over $\mathbb{F}_p$ that parametrizes smooth marked curves together with a non-zero differential form that is the differential of a meromorphic function. Furthermore, we consider the stack $\mathcal{M}_{g,n}^{ex, \mathbf{m}}$ that parametrizes those divisors on smooth curves that appear as divisors of exact differential forms. By introducing a local-global principle for first order deformations of the objects that we consider, we show smoothness of these stacks and compute their dimension.

math.AG↗

arXiv2026

Logarithmic Hurwitz Spaces in Mixed and Positive Characteristic with Wild Ramification

We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,\Xi}$ 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.

math.AG↗

arXiv2023

The moduli space of cyclic covers in positive characteristic

We study the $p$-rank stratification of the moduli space $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$, which represents $\mathbb{Z}/p^n$-covers in characteristic $p>0$ whose $\mathbb{Z}/p^i$-subcovers have conductor $d_i$. In particular, we identify the irreducible components of the moduli space and determine their dimensions. To achieve this, we analyze the ramification data of the represented curves and use it to classify all the irreducible components of the space. In addition, we provide a comprehensive list of pairs $(p,(d_1,d_2,\ldots,d_n))$ for which $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$ in characteristic $p$ is irreducible. Finally, we investigate the geometry of $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$ by studying the deformations of cyclic covers which vary the $p$-rank and the number of branch points.

math.AG↗