Stefan Schroeer
I show that the set of smooth curves of genus g admitting branched coverings X->P^1 with only triple ramification points has dimension at least max(2g-3,g). In characteristic two, such curves have tame rational functions and an analog of Belyi's Theorem applies to them.
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The uniform asymptotics for real double Hurwitz numbers with triple ramification I: the tropical correspondence
本文通过改进的热带对应关系，研究了具有三重分支的实双Hurwitz数的均匀渐进行为。
Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.
These are the notes accompanying 13 lectures given by the authors at the Clay Mathematics Institute Summer School 2014 in Madrid. The notes give an introduction into the theory of $\ell$-adic sheaves with emphasis on their ramification theory on curves.
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.
The uniform asymptotics for real double Hurwitz numbers with triple ramification II: lower bounds and asymptotics
本文研究了具有三重分支的实双Hurwitz数的均匀渐近行为，推导了其统一下界并证明了与复类似物的对数等价性。
We prove that in rank 1, the Abbes-Saito log conductor is determined by its restriction to curves. This result is essentially established by analyzing Artin-Schreier-Witt extensions. Consequently, we confirm an expectation of H. Esnault and M. Kerz. We also conjecture that this result holds in arbitrary finite rank. As an application, we translate recent results in higher class field theory of M. Kerz and S. Saito to the log-ramification context.
Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class of their compactifications in the moduli space of stable curves. Several results will be deduced from this computation.
In the 80's D. Eisenbud and J. Harris developed the general theory of limit linear series, Invent. math. 85 (1986), in order to understand what happens to linear systems and their ramification points on families of non-singular curves degenerating to curves of compact type. They applied their theory to the study of limits of Weierstrass points, among other endeavours. In one of their articles, Invent. math. 87 (1987), they asked: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" In this eprint we address this question within a …
We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.
In this article we study the modular properties of a family of cyclic coverings of P^1 of degree N, in all odd characteristics. We compute the moduli space of the corresponding algebraic stack over Z[1/2], as well as the Picard groups over algebraically closed fields. We put special emphasis on the study of the fibre of the stack at a prime of wild ramification; in particular we show that the moduli space has good reduction at such a prime.
The double ramification cycle satisfies a basic multiplicative relation DRC(a).DRC(b) = DRC(a).DRC(a + b) over the locus of compact-type curves, but this relation fails in the Chow ring of the moduli space of stable curves. We restore this relation over the moduli space of stable curves by introducing an extension of the double ramification cycle to the small b-Chow ring (the colimit of the Chow rings of all smooth blowups of the moduli space). We use this to give evidence for the conjectured equality between the (twisted) double ramification cycle and a cycle constructed in previous work by t…
Double ramification cycles on the moduli spaces of curves
本文研究了Deligne-Mumford稳定曲线模空间上的双重分歧循环，通过稳定映射模空间的虚拟基类定义，证明了Pixton在2014年猜想的显式公式，并给出了在Hodge积分计算中的应用。
Let C be a general connected, smooth, projective curve of positive genus g. For each nonnegative integer i we give formulas for the number of pairs (P,Q) em C x C off the diagonal such that (g+i-1)Q-(i+1)P is linearly equivalent to an effective divisor, and the number of pairs (P,Q) em C x C off the diagonal such that (g+i+1)Q-(i+1)P is linearly equivalent to a moving effective divisor.
We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain purel…
The moduli space of stable maps with divisible ramification uses $r$-th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer $r$. In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the $r$-ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.
The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.
Proves that if a curve has totally split reduction and the corresponding skeleton (as a metric graph) is a chain of loops with generic edge lengths, then every divisor on the graph with imposed ramification at the rightmost vertex $p$ of the skeleton lifts to a divisor class on the curve with the same ramification at some point $P$ that retracts to $p$, extending the work of Cartwright-Jensen-Payne. Gives a positive answer to lifting proper tropical intersections within a polytopal domain in an analytic torus.
For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditio…
Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.
It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general curves to the projective line, which even holds for wild ramification indices. Lastly, in the special case of rational functions on the projective line, we examine what we can say as a result about fami…
We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the g…
In this article, we prove that the Swan conductor of an étale sheaf on a smooth variety defined by Abbes and Saito's logarithmic ramification theory can be computed by its classical Swan conductors after restricting it to curves. It extends the same result for rank 1 sheaves due to Barrientos. As an application, we give a logarithmic ramification version of generalizations of Deligne and Laumon's lower semi-continuity property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.
Using the compactified universal jacobian over the moduli space of stable marked curves, we give an expression in terms of natural classes of the zero section of the compactified universal jacobian the (rational) Chow ring. After extending variants of the Abel-Jacobi map to a locus containing curves of treelike type we give a formula for the pullback of the said zero section along these extensions. The same approach is also applied to recover known formulas for the pullback of theta divisors to the moduli space of marked stable curves.
Let $k$ be a complete, nontrivially valued non-archimedean field. Given a finite morphism of quasi-smooth $k$-analytic curves that admit finite triangulations, we provide upper bounds for the number of connected components of the ramification locus in terms of topological invariants of the source curve such as its topological genus, the number of points in the boundary and the number of open ends.
We prove that $\bar {\mathbb Q}_\ell$-local systems of bounded rank and ramification on a smooth variety $X$ defined over an algebraically closed field $k$ of characteristic $p\neq \ell$ are tamified outside of codimension $2$ by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of $k$ of finite degree which fulfills the Lefschetz theorem. Last version: minor typos corrected.
Let C be a complete non-singular irreducible curve of genus 4 over an algebraically closed field of characteristic 0. We determine all possible Weierstrass semigroups of ramification points on double covers of C which have genus greater than 11. Moreover, we construct double covers with ramification points whose Weierstrass semigroups are the possible ones.
We show that three numerical semigroups <5,6,7,8>, <3,7,8 > and <3,5> are of double covering type, i.e., the Weierstrass semigroups of ramification points on double covers of curves. Combining this with the results of Oliveira-Pimentel and Komeda we can determine the Weierstrass semigroups of the ramification points on double covers of genus two curves.
Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces
研究了具有指定分支的圆柱面在曲线模空间和$K3$表面塞维里品种上的Brill-Noether定位，并探讨了其在$K3$表面对应关系和循环中的应用。
We study branched covers of curves with specified ramification points, under a notion of equivalence derived from linear series. In characteristic 0, no non-constant families of covers with fixed ramification points exist. In positive characteristic we formulate a necessary and sufficient condition for the existence of such a family. We unconditionally prove one direction of this conjecture, and by studying infinitesimal deformations show the other direction in characteristic 2 and 3.
Integrals of psi-classes over double ramification cycles
针对曲线模空间上DR-循环尚无标准tautological类表达式的问题，本文计算了零维交点情形下DR-循环与ψ类任意单项式的相交数。
We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts und…
A correlated refinement of the double double ramification cycle
本文通过根部的Weil配对，对双重重叠分支循环进行了细化，证明了细化类满足多重覆盖公式，并应用于环面曲面的对数Gromov-Witten不变量，得到了N. Takahashi猜想的变体。
In this paper, using the formula for the integrals of the $ψ$-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the $n$-point function of the intersection numbers on the moduli space of curves.
In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germ of curves over a complete discrete valuation ring of inequal characteristics (0,p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-Sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p…
Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a `universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
Cusps in $\mathbb{C}^3$ with prescribed ramification
本文研究具有固定分歧轮廓的映射芽的值半群，并证明在特定条件下，$\mathbb{P}^3$ 中单尖有理曲线的 Severi 簇常可约，并揭示了与加法组合学和数论的新联系。
Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $φ$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that is ramified over a smooth curve of degree 6. The surface $X$ contains 240 curves with negative self-intersection, called exceptional curves. We prove that for a point~$P$ on the ramification curve of $φ$, at most sixteen exceptional curves go through~$P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, we prove that for a p…
An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called restrained, if all second ramification groups are trivial (e.g., every group action on an ordinary curve is restrained). When the ramification indices satisfy certain numerical criteria, we construct a degenerating equivariant quasi-projective family to which the given curve belongs, and which in a sense is the unique building block for all such restrained equivariant families that ramify above a fixed set of points. The result is used to inductively study automorphi…
We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is…
We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed smooth proper source curve $X$. We show such a family exists precisely when the following conditions are satisfied: $\operatorname{char}(k)=p>0$, $X$ is isomorphic to $\mathbb{P}^1_k$, there is a unique ramification point, and the Galois group is $(\mathbb{Z}/p\mathbb{Z})^m$ for some integer $m>0$.
Rooted trees with level structures, $Ω$-classes and double ramification cycles
本文在曲线模空间重言环中证明了一类涉及带层级结构有根树、霍奇丛顶部陈类、Ω-类及双分歧结构的新关系系统，解决了相关猜想，并与DR/DZ等价性建立联系。
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.
Given a generically étale morphism $f\colon Y\to X$ of quasi-smooth Berkovich curves, we define a different function $δ_f\colon Y\to[0,1]$ that measures the wildness of the topological ramification locus of $f$. This provides a new invariant for studying $f$, which cannot be obtained by the usual reduction techniques. We prove that $δ_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula, and show that $δ_f$ can be used to explicitly construct the simultaneous skeletons of $X$ and $Y$. As an application, we use …
Pluricanonical cycles and tropical covers
本文从多重典范双分歧环的对数相交理论中提取新数值不变量，证明其可由修正平衡条件的热带曲线计数计算、为分段多项式且是 Fock 空间算子矩阵元，并将传统双 Hurwitz 数推广到多重典范情形。
The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.
Triangulations of non-archimedean curves, semi-stable reduction, and ramification
本文利用 Berkovich 曲线的最小三角剖分研究非阿基米德曲线的半稳定约化，证明最小三角剖分重数的最小公倍数整除 Galois 扩张次数，并在驯顺情形下给出 Saito 定理的新证明及椭圆曲线和野分歧情形的推广分析。
We prove an A'Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field $K$. As a first application, we relate the orders of the tame monodromy eigenvalues on the $\ell$-adic cohomology of a $K$-curve to the geometry of a relatively minimal $sncd$-model, and we show that the semi-stable reduction theorem and Saito's criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper $K$-varieties. We see that the validity of the t…
Reverse Hurwitz counts of genus 1 curves
本文研究亏格1曲线的反向Hurwitz计数问题，建立了特定非简单无标记分歧剖面下的封闭形式结果与递推公式，并证明某特殊情形的解为关于$d$的显式四次多项式。
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.
This paper is an extended version of the author's talk given at the conference "Non-Archimedean analytic geometry: theory and practice" held in August 2015 at Papeete. It gives a brief overview of recent results on the structure of wild coverings of Berkovich curves and its relation to the different and higher ramification theory.
Logarithmic double ramification cycles
本文给出了对数双分歧环的显式公式，通过研究曲线模空间上的泛 Jacobian 并应用 Holmes-Schwarz 判据，将 Pixton 公式提升到对数 Chow 环，并计算了若干例子。
We develop a theory for stable maps to curves with divisible ramification. For a fixed integer $r>0$, we show that the condition of every ramification locus being divisible by $r$ is equivalent to the existence of an $r$th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to $r$-s…
This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring-Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic $p>2$.
A point of an algebraic curve of genus g is subcanonical if some regular differential vanishes only at that point, with multiplicity 2g-2. Subcanonical points are Weierstrass points, and we compute the associated gap sequence at a general point of each component of the moduli space of curves with marked subcanonical point. We also construct subcanonical points with other gap sequences as ramification points of certain cyclic covers.
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
Non-cycle triple planes with branch curve of degree at most 10
本文研究了分支曲线次数不超过10的正常非循环三重覆盖的分类问题。
The Hodge-Double-Ramification conjecture and Mumford's formula on the universal Picard stack
本文证明了霍奇-双分歧猜想及对数DR圈的类似猜想，前者归约为已有局部化计算，后者基于Thom--Porteous公式，还推广了泛线丛陈特征的芒福德公式。
Let X be a smooth projective curve over a field of characteristic p>0 and G a finite group of automorphism of X. Let n(X,G) be the characteristic of the versal equivariant deformation ring R(X,G) of (X,G). When the ramification is weak (i.e. all second ramification groups are trivial),we prove that n(X,G) is 0 or p and we compute R(X,G).
We apply the theory of the radius of convergence of a p-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. In the case of an etale covering of curves with good reduction, we get a lower bound for that radius and obtain a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich. We take this opportunity to clarify the relation between our notion of radius of convergencand the more intrinsic one used by Kedlaya
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