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We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with…
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…
We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group…
Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz…
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…
Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial…
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called…
We describe double Hurwitz numbers as intersection numbers on the moduli space of curves. Assuming polynomiality of the Double Ramification Cycle (which is known in genera 0 and 1), our formula explains the polynomiality in chambers 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…
Double Hurwitz numbers count branched covers of the projective line with fixed branch points, with simple branching required over all but two points 0 and infinity, and the branching over 0 and infinity specified by partitions of the degree…
We present the multi-matrix models that are the generating functions for 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 genus,…
We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation…
We give an alternative proof of the Hurwitz existence problem for branched covers of $\mathbb{P}^1$ in the case where the number of ramification points equals the number of branch points, that is, where all the ramification profiles are of…
We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten…
We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show…
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.…
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…
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 continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate…
We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only…