Hurwitz Spaces Research Articles

Polynomial stability of the homology of Hurwitz spaces

Polynomial stability of the homology of Hurwitz spaces

Low-degree Hurwitz stacks in the Grothendieck ring

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 this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.

MFO–RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory

This report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck’s original pillars [1], the present program has now evolved beyond the classical group-theoretic legacy to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry . This panorama “closes the loop” by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara’s program and Nakamura–Tamagawa–Mochizuki’s anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations .This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers. [1] That are, as presented in “Récoltes et semailles”, the resolution of the discrete and the continuous, the local-to-global thinking by generization-specialization, and the quintessential intersection of arithmetic and geometry – see § 2.10 ibid.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.

Generic Beauville’s Conjecture

Abstract Let $\alpha \colon X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha $ is semistable if the genus of Y is at least $1$ and stable if the genus of Y is at least $2$ . We prove this conjecture if the map $\alpha $ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve Y.

Meromorphic functions without real critical values and related braids

We study the open subset of the Hurwitz space, consisting of meromorphic functions of a given degree defined on closed Riemann surfaces of a given genus and having no real critical values, and enumerate its connected components in terms of braids. Specifically, to a function in this open set, we assign a braid in the braid group of the underlying closed surface and characterize all braids which might appear using this construction. We introduce the equivalence relation among these braids such that the braids corresponding to the meromorphic functions from the same connected component of the above Hurwitz space are equivalent while non-equivalent braids correspond to distinct connected components. Several special families of meromorphic functions, some applications, and further problems are discussed.

Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space Mg,n of Riemann surfaces of genus g≥0 with n≥1 ordered and directed marked points. For d≥2g+n−1 we show that Mg,n is homotopy equivalent to a component of the simplicial Hurwitz space HurΔ(Sdgeo) associated with the partially multiplicative quandle Sdgeo. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space Ω∞−2MTSO(2) of Hurwitz flavour.

The Hurwitz space Picard rank conjecture for d > g − 1

We show the simple Hurwitz space Hg,d has trivial rational Picard group for d>g−1.

Hurwitz–Ran spaces

Given a couple of subspaces {mathcal {Y}}subset {mathcal {X}} of the complex plane {mathbb {C}} satisfying some mild conditions (a “nice couple”), and given a PMQ-pair ({mathcal {Q}},G), consisting of a partially multiplicative quandle (PMQ) {mathcal {Q}} and a group G, we introduce a “Hurwitz–Ran” space text {Hur}({mathcal {X}},{mathcal {Y}};{mathcal {Q}},G), containing configurations of points in {mathcal {X}}setminus {mathcal {Y}} and in {mathcal {Y}} with monodromies in {mathcal {Q}} and in G, respectively. We further introduce a notion of morphisms between nice couples, and prove that Hurwitz–Ran spaces are functorial both in the nice couple and in the PMQ-group pair. For a locally finite PMQ {mathcal {Q}} we prove a homeomorphism between text {Hur} ((0,1)^2;{mathcal {Q}}_+) and the simplicial Hurwitz space text {Hur} ^{Delta }({mathcal {Q}}), introduced in previous work of the author: this provides in particular text {Hur} ((0,1)^2;{mathcal {Q}}_+) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.

Gromov-Witten invariants of 𝑆𝑦𝑚^{𝑑}ℙ^{𝕣}

We give a graph-sum algorithm that expresses any genus- g g Gromov-Witten invariant of the symmetric product orbifold Sym d ⁡ P r ≔ \operatorname {Sym}^d\mathbb {P}^r≔ [ ( P r ) d / S d ] [(\mathbb {P}^r)^d/S_d] in terms of “Hurwitz-Hodge integrals” – integrals over (compactified) Hurwitz spaces. We apply the algorithm to prove a mirror-type theorem for Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r in genus zero. The theorem states that a generating function of Gromov-Witten invariants of Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r is equal to an explicit power series I Sym d ⁡ P r I_{\operatorname {Sym}^d\mathbb {P}^r} , conditional upon a conjectural combinatorial identity. This is a first step in the direction of proving Ruan’s Crepant Resolution Conjecture for the resolution H i l b ( d ) ( P 2 ) Hilb^{(d)}(\mathbb {P}^2) of the coarse moduli space of Sym d ⁡ P 2 . \operatorname {Sym}^d\mathbb {P}^2.

The Chow rings of the moduli spaces of curves of genus 7, 8, and 9

The rational Chow ring of the moduli space M g \mathcal {M}_g of curves of genus g g is known for g ≤ 6 g \leq 6 . Here, we determine the rational Chow rings of M 7 , M 8 \mathcal {M}_7, \mathcal {M}_8 , and M 9 \mathcal {M}_9 by showing they are tautological. The key ingredient is intersection theory on Hurwitz spaces of degree 4 4 and 5 5 covers of P 1 \mathbb {P}^1 via their associated vector bundles. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on P 1 \mathbb {P}^1 are highly unbalanced, expanding upon previous work of the authors in the more balanced case. In genus 9 9 , we use work of Mukai to present the locus of hexagonal curves as a global quotient stack, and, using equivariant intersection theory, we show its Chow ring is generated by restrictions of tautological classes.

Low-dimensional bihamiltonian structures of topological type

We construct local bihamiltonian structures from classical W-algebras associated with non-regular nilpotent elements of regular semisimple type in Lie algebras of types A2 and A3. They form exact Poisson pencils and admit a dispersionless limit, and their leading terms define logarithmic or trivial Dubrovin–Frobenius manifolds. We calculate the corresponding central invariants, which are expected to be constants. In particular, we get Dubrovin–Frobenius manifolds associated with the focused Schrödinger equation and Hurwitz space M0;1,0 and the corresponding bihamiltonian structures of topological type.

The integral Picard groups of low-degree Hurwitz spaces

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 simply branched covers. For simple branching, the Picard groups are finite, with order depending on the genus.

A transfer principle: from periods to isoperiodic foliations

We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus $$g\ge 2$$ curves and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne–Mumford compactification of the moduli space of curves.

On generators and relations of the rational cohomology of Hilbert schemes

We consider for $$d\ge 1$$ the graded commutative $${\mathbb {Q}}$$ -algebra $${\mathcal {A}}(d):=H^*({\text {Hilb}}^d({\mathbb {C}}^2);{\mathbb {Q}})$$ , which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $${\lfloor }d/2{\rfloor }$$ multiplicative generators of $${\mathcal {A}}(d)$$ . Additionally, we prove when the lowest degree generating relations occur. For small values of d, we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $${\mathcal {A}}(d)$$ .

Tautological classes on low-degree Hurwitz spaces

Abstract Let ${\mathcal {H}}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of ${\mathbb {P}}^{1}$. We define the tautological ring of ${\mathcal {H}}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of “factoring covers,” are tautological up to codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also used in our subsequent work [6], wherein we prove new results about the structure of the Chow ring for $k \leq 5$.

Generalized Tevelev degrees of [formula omitted

Let (C,p1,…,pn) be a general curve. We consider the problem of enumerating covers of the projective line by C subject to incidence conditions at the marked points. These counts have been obtained by the first named author with Pandharipande and Schmitt via intersection theory on Hurwitz spaces and by the second named author with Farkas via limit linear series. In this paper, we build on these two approaches to generalize these counts to the situation where the covers are constrained to have arbitrary ramification profiles: that is, additional ramification conditions are imposed at the marked points, and some collections of marked points are constrained to have equal image.

Equations of linear subvarieties of strata of differentials

For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere.

Genus zero of projective symplectic groups

A transitive subgroup G ≤ SN is called a genus zero group if there exist non identity elements x1 , . . . , xr∈G satisfying G =<x1, . . . , xr>, x1·...·xr=1 and ind x1+...+ind xr = 2N − 2. The Hurwitz space Hinr(G) is the space of genus zero coverings of the Riemann sphere P1 with r branch points and the monodromy group G.In this paper, we assume that G is a finite group with PSp(4, q) ≤ G ≤ Aut(PSp(4, q)) and G acts on the projective points of 3-dimensional projective geometry PG(3, q), q is a prime power. We show that G possesses no genus zero group if q > 5. Furthermore, we study the connectedness of the Hurwitz space Hinr(G) for a given group G and q ≤ 5.