L. Bonavero, C. Casagrande, S. Druel
We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety of dimension at most 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone.
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In this paper we address Fano manifolds X with a locally unsplit dominating family of rational curves of anticanonical degree equal to the dimension of X. We first observe that their Picard number is at most 3, and then we provide a classification of all cases with maximal Picard number. We also give examples of locally unsplit dominating families of rational curves whose varieties of minimal tangents at a general point is singular.
We study Fano manifolds $X$ admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if $X$ has pseudoindex $i_X = (\dim X)/3$ or dimension $\dim X=6$. We also show that this conjecture is true for all Fano manifolds with $i_X > (\dim X)/3$.
As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that $X$ is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some …
Rational Exceptional Belyi Coverings
本文研究亏格0紧Riemann面的有理例外Belyi覆盖，利用Maple识别了度数至15的所有此类覆盖，计算了部分Belyi函数，并发现了新的无穷级数。
In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.
We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.
The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …
We classify purely inseparable morphisms of degree $p$ between rational double points (RDPs) in characteristic $p > 0$. Using such morphisms, we refine a result of Artin that any RDP admits a finite smooth covering.
Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the double covering of projective space of dimension 6, 7, 8 or 9, branched along a very general quartic hypersurface. On the other hand, such double coverings as well as general quartic hypersurfaces of dimension at least 5 are known to be unirational.
We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.
Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $π: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+kπ^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
We prove that a double covering of P^3 branched along a very general sextic surface is not stably rational.
Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,\tilde{g}:\mathbb{A}^2 --\to S$, such that the union of their images covers the affine surface $S\cap \mathbb{A}^n$. In the affine case, the number of rational maps involved in the cover is in general…
We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.
The number A(q) is the upper limit of the ratio of the maximum number of points of a curve defined over $\Fq$ to the genus. By constructing class field towers with good parameters we present improvements of lower bounds of A(q) for q an odd power of a prime.
Let X be a Mori dream space together with an effective torus action of complexity one. In this note, we construct a polyhedral divisor on a suitable covering of the projective line P^1 which corresponds to the affine spectrum of the Cox ring of X. This description allows for a detailed study of torus orbits and deformations of the latter. Moreover, we present coverings of P^1 together with an action of a finite abelian group A in terms of so-called A-divisors of degree zero on P^1.
This paper deals with a study of the rational elliptic surfaces whose $J$-invariant functions are of degree one. Almost all of these elliptic surfaces have four singular fibers, while the remaining surfaces have only three singular fibers. The moduli space of these elliptic surfaces is canonically isomorphic to the projective line by taking the $J$-values for a certain fixed type of singular fibers. Over the moduli space, we discuss our elliptic surfaces, and investigate how their sections are described by the parameter of the moduli space. By using a covering space of the moduli, we construct…
An Effective Criterion for Covering Maps Between Real Varieties
本文提出了一种实变种间覆盖映射的有效判定准则，通过算法化方法验证覆盖性质。
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
Enriques surfaces with normal K3-like coverings
本文针对特征2下类K3覆盖为正规的单连通Enriques曲面，基于翻转变换Frobenius拉回的构造方法，结合多项代数几何经典结果，分析其结构并建立相关奇点、切层与典范覆盖理论。
A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…
We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill-Noether Theory for pencils on real curves. For coverings having topological degree 0 we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.
A class of rational functions characterized by some wonderful properties is studied. The properties that identify this class include simple algebra (their inverses can be expressed in radicals), simple topology (the total space of the minimal Galois covering dominating them has genus 0 or 1) and simple local topol- ogy (branching data). Explicit formulae for these functions are obtained as well as their classification up to different equivalence relations.
Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.
Slope-determinant method, complex cellular structures and hypersurface coverings of regular rational points
本文结合 Bombieri-Pila/Heath-Brown 行列式方法、Chen 的 Arakelov 斜率重构与 Binyamini-Novikov 复细胞结构，将正则有理点超曲面覆盖计数中的次多项式因子改进为多对数因子。
We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves.
In this paper we study smooth complex projective polarized varieties (X,H) of dimension n \ge 2 which admit a dominating family V of rational curves of H-degree 3, such that two general points of X may be joined by a curve parametrized by V, and such that there is a covering family of rational curves of H-degree one. Our main result is that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjuction theoretic scroll structure over a smooth variety.
Cyclic coverings of rational normal surfaces which are quotients of a product of curves
本文研究作为曲线乘积之商的有理正规曲面的循环覆盖，推广 Esnault-Viehweg 方法证明单值群在第一 Betti 群及 Hodge 结构上的作用可分解为 P^1 上特定循环覆盖的直和，并应用于 Lê-Yomdin 曲面奇点和等平凡纤维化曲面。
In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has a vanishing higher cohomology, \begin{equation} H^1(N_{c_0/X_0})=0. \end{equation} As applications we give (2) A solution to a Voisin's conjecture [9] on a covering of a generic hypersurface by rational curves (3) A classification of rational curves on hypersurfaces of general type--a solution to another Voisin's conjecture [9].
Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibr…
We study smooth complex projective polarized varieties $(X,H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by $V$ and which do not admit a covering family of lines (i.e. rational curves of $H$-degree one). We prove that such manifolds are obtained from RCC manifolds of Picard number one by blow-ups along smooth centers. If we further assume that $X$ is a Fano manifold, we obtain a stronger result, classifying all Fano RCC manifolds of Picard number $ρ_X \ge 3$
We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^…
Iterated Grassmannian bundles over moduli stacks of vector bundles on a curve are shown to be birational to an affine space times a moduli stack of degree 0 vector bundles, following the method of King and Schofield. Applications include the birational type of some Brill-Noether loci, of moduli schemes for vector bundles with parabolic structure or with level structure and for A. Schmitt's decorated vector bundles. A further consequence concerns the existence of Poincare families on finite coverings of the moduli schemes.
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness via the $\mathbb{A}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all $\mathbb{A}^1$-curves. As an …
Let $ϕ: X \to \mathbb P^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \leq m \leq n-1$. We study the varieties of minimal rational tangents $\mathcal C_x \subset \mathbb P T_x(X)$ at a general point $x$ of $X$. We describe the homogeneous ideal of $\mathcal C_x$ and show that the projective isomorphism type of $\mathcal C_x$ varies in a maximal way as $x$ varies over general points of $X$. Our description of the ideal of $\mathbb C_x$ implies a certain rigidity property of the covering morphism $ϕ$. As an application of this rigidity, we show that any finite morphi…
Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.
We study cyclic finite Galois extensions of the rational function field of the projective line P^{1}(F_q) over a finite field F_q with q elements defined by considering quotient curves by finite subgroups of the projective linear group PGL(2,q), and we enumerate them expressing the count in terms of Stirling numbers.
We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if X is a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then any dominant rational mapping from X to projective n-space must have degree at least d-1. We also propose a number of open problems,…
In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to …
Nodal rational curves on Enriques surfaces of base change type
本文提出一种纯几何无格构造方法，证明了基变换型Enriques曲面上双截面通常是节点的，并展示了此类曲面承认形成秩8子群的节点有理曲线集。
We give another solution to the class number one problem by showing that imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve $\mscrK_3$. In fact one can find all rational points on $\mscrK_3$. The curve $\mscrK_3$ arises naturally via certain coverings of curves:\ $\mscrK_3\rg\mscrK_6$,\ $\mscrK_1\rg\mscrK_2$\ with $\mscrK_2\colon y^2=2x(x^3-1)$ denoting the Heegner curve, also in connection with the so-called Heegner-Stark covering $\mscrK_1\rg\mscrK_s$.
Kummer surfaces associated with group schemes
本研究在特征2情形下引入由群概形mu_2作用于有理尖点曲线自乘积得到的Kummer曲面，证明其恰为Artin不变量σ≤3的超奇异K3曲面，给出曲线构型刻画并说明其可作为单连通Enriques曲面的正规K3型覆盖。
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.
On algebraic and non-algebraic neighborhoods of rational curves
本文研究有理曲线邻域的代数性问题，构造了自交数任意正整数d的ℙ¹非代数嵌入芽，其亚纯函数芽域超越次数为2，并给出两种构造及基于ℙ¹嵌入代数芽分类的非代数性证明。
What Kind of Morphisms Induces Covering Maps over a Real Closed Field?
本文研究平坦态射在实闭域上诱导覆叠映射的条件，证明约化后为有限平展并在有理点上诱导覆叠映射，并应用于柱面代数分解。
We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms.
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.
In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied by Poon and Kreussler-Kurke to the case of arbitrary signature. In particular, the branch divisor of the double covering is a cut of the rational threefold by a single quartic hypersurface. A defining equation of the hypersurface is determined in an explicit form. We also show that these twistor spaces interpolate LeBrun twistor spaces and the twistor spaces constructed in mat…
Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to the projective line whose Galois group is solvable. We give an example of a genus 7 complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z' -> Z with Z' rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci W_d(C) in the Jacobian of a curve C may contain translates of abelian subvarie…
We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a com…
Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points
本文研究了超椭圆曲线的动机上同调中的调节器构造，并通过超几何函数验证了其非平凡性及整数性，同时探讨了 $L$-函数特殊值的贝利松猜想。
Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…
For a semifield extension $T /S$, an action of a finite group $G$ on $T$ is Galois if $(1)$ the $G$-invariant subsemifield of $T$ is $S$ and $(2)$ subgroups of $G$ whose invariant semifields coincide are equal. We show that for a finite harmonic morphism between tropical curves $φ: \varGamma \to \varGamma^{\prime}$ and an isometric action of a finite group $G$ on $\varGamma$, $φ$ is $G$-Galois if and only if the natural action of $G$ on the rational function semifield $\operatorname{Rat}(\varGamma)$ of $\varGamma$ induced by the action of $G$ on $\varGamma$ is Galois for the semifield extensio…
To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any completion of Q but no model over Q.
A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then X admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree 1. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.
We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with larg…
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of Gromov-Witten invariants for all algebraic projective or closed symplectic manifolds. Mirror Symmetry in the basic example of rational curves on a quintic 3-folds is reduced to certain complicated but explicit identity. The strategy of computations can be described as follows: 1) we reduce counting problems to questions concerning Chern classes on spaces of curves on…
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