Related papers: Solution of the Hurwitz problem with a length-2 pa…

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of…

We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in…

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted…

Hirschfeld classified split del Pezzo surfaces of degree at least three whose points are all contained on the lines in the surface. We continue his work and begin the classification of split degree two del Pezzo surfaces over finite fields…

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a…

The Chisini conjecture asserts that a generic ramified covering over the complex projective plane of degree at least 5 is uniquely determined by its branch curve. We prove this for degree at least 12 using the work of Kulikov…

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb P^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify…

Hurwitz numbers enumerate branched morphisms between Riemannn surfaces with fixed numerical data. They represent important objects in enumerative geometry that are accessible by combinatorial techniques. In the past decade, many variants of…

Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating…

In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

We define a new class of enumerative invariants called $k$-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the $k$-leaky double Hurwitz numbers introduced in previous work of…

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a…

In [14] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some (2,1^{d-2}) profiles. In this paper, motivated from [3], we generalize these results to Hurwitz numbers…

E. Kani has shown that the Hurwitz functor, which parametrizes the (normalized) genus 2 covers of degree 3 of an elliptic curve, is representable. In this paper the corresponding moduli scheme and the universal family are explicitly…

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of…

For the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts, we develop a degree theory with three levels: exact value theory, exact profile theory, and…

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p…

We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain…

We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of…

Hurwitz numbers with completed cycles are standard Hurwitz numbers with simple branch points replaced by completed cycles. In fact, simple branch points correspond to completed $2$-cycles. Okounkov and Pandharipande have established the…