Related papers: Covering semigroups
In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target -- $\mathbb P^1$ with a list of $n$ points $q_1,\dots,q_n\in \mathbb P^1$ -- and partitions…
The {\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\rm S} \subset…
We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underline{\mu}))$ vanishes for covers of degree $ d = 3$ and deduce the same result…
Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. Given a $\mathbb{Z}/p$-Galois cover of a formal disc over $R$, one can derive from it a semi-stable model for which the specializations of branch points are…
The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification…
We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions…
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption…
In this paper, we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p. Our techniques involve related computations in the case of three branch…
Let $\varphi:\Sigma_1\longrightarrow \mathbb{P}^2$ be a blow up at a point on $\mathbb{P}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\geq 4$ by $\varphi$, and let $P$ be a point on $C$. Let…
When the Seiberg-Witten curve of a four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory wraps a Riemann surface as a multi-sheeted cover, a topological constraint requires that in general the curve should develop ramification…
Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$.…
In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d-sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the…
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to…
We compute the Picard groups with integral coefficients of the Hurwitz stacks parametrizing degree $4$ and $5$ covers of $\mathbb{P}^1$. As a consequence, we also determine the integral Picard groups of the Hurwitz stacks parametrizing…
For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has…
We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…
We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed…
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…
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points…