Related papers: $k$-leaky double Hurwitz descendants

Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz…

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…

The Hurwitz form of a variety is the discriminant that characterizes linear spaces of complementary dimension which intersect the variety in fewer than degree many points. We study computational aspects of the Hurwitz form, relate this to…

We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the…

We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone…

Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of…

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show…

We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We…

We consider special series in ratios of the Schur functions which are defined by integers $\textsc{f}\ge 0$ and $\textsc{e} \le 2$, and also by the set of $3k$ parameters $n_i,q_i,t_i,\,i=1,..., k$. These series may be presented in form of…

Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of…

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios…

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many…

We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this…

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant…

We construct refined tropical enumerative genus zero invariants of toric surfaces that specialize to the tropical descendant genus zero invariants introduced by Markwig and Rau when the quantum parameter tends to $1$. In the case of…

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety.…

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the…

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of…

We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $\beta$-Grothendieck polynomials…

We study the density of the invariant measure of the Hurwitz complex continued fraction from a computational perspective. It is known that this density is piece-wise real-analytic and so we provide a method for calculating the Taylor…