M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.
This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several res…
We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at the point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale s…
Constructible complexes have the same characteristic cycle if they have the same wild ramification, even if the characteristics of the coefficients fields are different.
We introduce the notion that two elements of Grothendieck groups of constructible sheaves on a separated scheme over an excellent henselian discrete valuation ring have the same wild ramification. We prove that this condition is preserved by four of Grothendieck's six operations except the derived tensor product and the derived hom.
We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler-Poincaré characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field.
We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.
In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers.
Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature insteresting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in \cite{DN13}. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy…
For $i\geq2$, we compute the first coefficients of the class $[\overline{D}(μ;3)]$ in the rational Picard group of the moduli of Prym curves $\overline{\mathcal{R}}_{2i}$, where $D(μ;3)$ is the divisor parametrizing pairs $[C,η]$ for which there exists a degree $2i$ map $π\colon C\rightarrow \mathbb{P}^1$ having ramification profile $(2,\ldots,2)$ above two points $q_1, q_2$, a triple ramification somewhere else and satisfying $\mathcal{O}_C(\frac{π^{*}(q_1)-π^{*}(q_2)}{2})\cong η$. Furthermore, we provide several new Prym enumerative results related to this situation.
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. Our definition of ramification is relative to an algebraic family of divisors on the target variety, and our formula is obtained using the theory of refined top Chern classes. In assigning multiplicities to ramification points, we frequently have to consider excess degeneracy loci, but we are able to show nonetheless that the multiplicities are always nonnega…
We define the notion of inseparable coverings of schemes and we propose a ramification formalism for them, along the lines of the classical one. Using this formalism we prove a formula analogous to the classical Riemann-Hurwitz formula for generic torsors under infinitesimal diagonalizable group schemes.
We prove that every connected affine scheme of positive characteristic is a K(pi, 1) space for the etale topology. The main ingredient is the special case of the affine space over a field k. This is dealt with by induction on n, using a key "Bertini-type"' statement regarding the wild ramification of l-adic local systems on affine spaces, which might be of independent interest. Its proof uses in an essential way recent advances in higher ramification theory due to T. Saito. We also give rigid analytic and mixed characteristic versions of the main result.
Università degli studi di Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所（INFN）的里雅斯特分部）；Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院）
本文通过基本方法研究Hurwitz数在高 genus 下的渐进行为，涵盖单、双及任意固定分支类型，以及简单或包含完整循环等分支类型，总结了主要贡献。
A finite étale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem…
The \(v\)-function appears in the wild McKay correspondence and it is an invariant of a Galois extension over a local field which measures about the ramification of extension. The ramification filtration is also such invariant. Yasuda presented the problem whether the \(v\)-function is determined by ramification filtration. In this paper, we give a formula for the \(v\)-function for 2-dimensional representations of the bicyclic group of order \(p^2\) with \(p\) the characteristic of our base field. Using this formula, we construct an example of \(v\)-function not determined by the ramification…
In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito's logarithmic ramification filtration. This provides a positive answer to the main conjecture in Isabel Leal's article "On the ramification of étale cohomology groups" for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.
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 monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramif…
The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of $\mathbb{P}^1$ with a fixed ramification profile over infinity. We prove that primitive forms are in one-to-one correspondence with semi-simple Frobenius structures on the base of the family. Furthermore, we introduce the notion of a polynomial primitive form and show that the corresponding class of Frobenius manifolds contains the Hurwitz Frobenius manifolds of Dubrovin. Finally, we apply our theory to investigate the relation between the Eynard--Orantin recursion and Frobenius…
We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $ε$ and a quantum parameter $\hbar$. When $\hbar=0$, this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the $ε=0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent h…
Given a branched cover $f:Y\to X$ between smooth projective curves over a non-archimedian mixed-characteristic local field and an open rigid disk $D\subset X$, we study the question under which conditions the inverse image $f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen.
The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.
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 genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. Ramifications at other $n-2$ points enter the sum with the length of the profile at $z_2$ and with the total length of profiles at the remaining $n-3$ points. We find the spectral curve of the model for $n=5$ using the loop equation technique for the above generating function represente…
In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin …
For any nonconstant f,g in C(x) such that the numerator H(x,y) of f(x)-g(y) is irreducible, we compute the genus of the normalization of the curve H(x,y)=0. We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves f:A-->D and g:B-->D.
