Luis Fuentes Garcia
Let X be a smooth curve of genus g. When pi>=3g and d>=pi-2g+1 we show the existence of a double covering gamma:C-->X where C a smooth curve of genus pi with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose, we generalize the classical results of Maroni concerning base-point-free pencils on trigonal curves to the case of triple covers of arbitrary smooth irrational curves.
We prove that a general Fano fibration $π\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.
We complete the study of birational geometry of Fano fiber spaces $π\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces $V/{\mathbb P}^1$ is developed.
We continue to study birational geometry of Fano fibrations $π\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of sufficient twistedness over the base, we prove birational rigidity (in particular, it means that there are no other structures of a fibration into rationally connected varieties) and compute their groups of birational self-maps. We considerably improve the principal components of the method of maximal singularities, in the first place, the technique of counting multip…
Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.
We show that the strata $ \mathcal{M}_{11,6}(k) \subset \mathcal{M}_{11} $ of $ 6-$gonal curves of genus $ 11 $, equipped with $k$ mutually independent and type I pencils of degree six, have a unirational irreducible component for $5\leq k\leq 9$. The unirational families arise from degree $ 9 $ plane curves with $ 4 $ ordinary triple and $ 5 $ ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree $ 8 $ plane curves with $ 10 $ ordinary double points covers an irreducible component of excess dimension in $ \mathcal{M…
In the moduli space $\mathcal{R}_g$ of double étale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors $\mathcal{T}^e_g$ and $\mathcal{T}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne-Mumford compactification $\overline{\mathcal{R}}_g$.
Let $C$ be a smooth curve with gonality $k\ge 6$ and genus $g\ge 2k^2+5k-6$. We prove that $W^1_d({C})$ has the expected dimension and that the general element of any irreducible component of $W^1_d({C})$ is primitive if either $g-k+4\le d\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k/2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g^1_{g-k+3}$.
We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.
We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of ration…
Geometry of Prym semicanonical pencils and an application to cubic threefolds
本研究探讨模空间中半典范线束对应除子上的Prym映射，揭示其差异与低亏格几何特性，并通过对特定除子的分析得到三次三维簇直线的计数结果。
Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities
本文确立了具有半齐次奇点的射影超曲面束的中间维上同调的稳定 Griffiths 高度公式，通过构造有限覆盖以获得半稳定光滑模型，成功规避了确定奇点纤维初等指数的难题。
In this paper we discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumption, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obta…
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and computations. We also discuss about the sharpness of our main result and the so-called Castelnuovo-Severi bound byexhibiting some examples.
We prove the generic injectivity of the Prym map, sending a double covering of an elliptic curve ramified at r>4 points to its polarized Prym variety. For r=6 the map is birational.
This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when the Picard number is one. It turns out that all of them are new examples.
We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the stack of admissible double coverings.
In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painlevé I equation. We will show that this system passes the Painlevé $α$-test for all accessible singular points $P_i \ (i=1,2,3)$. We note its holomorphy condition of the first Painlevé system.
Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves
本文研究了具有p_g=K^2=1的曲面中曲率2 pencil的存在性，证明其在模空间中形成不可约子流形，并探讨了与平面三次曲线相切的圆锥线几何关系。
The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are…
In this paper we construct 206 examples of Calabi-Yau manifolds with different Euler numbers. All constructed examples are smooth models of double coverings of $P^3$ branched along an octic surface. We allow 11 types of (not necessary isolated) singularities in the branch locus. Thus we broaden the class of examples studied in math.AG/9902057. For every considered example we compute the Euler number and give a precise description of a resolution of singularities.
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 $S$.
We prove that the ramified Prym map $\mathcal P_{g, r}$ which sends a covering $π:D\longrightarrow C$ ramified in $r$ points to the Prym variety $P(π):=\text{Ker}(\text{Nm}_π)$ is an embedding for all $r\ge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that $\mathcal P_{g, 2}$ and $\mathcal P_{g, 4}$ have positive dimensional fibers.
Prym loci of branched double coverings and generalized Andreotti-Mayer loci
本文将安德烈奥蒂-迈耶轨迹推广至非主极化阿贝尔簇模空间，证明分支二重覆盖的普里姆轨迹是广义安德烈奥蒂-迈耶轨迹的不可约分支。
Let E be a rank two vector bundle on a scheme X. The following three structures are shown to be equivalent : a) A primitive quadratic map q: E --> L, with values in an invertible module L. b) A double covering f: Y --> X endowed with an invertible module F on Y, plus an isomorphism from the direct image of F to E. c) An effective Cartier divisor on the projective space P(E), of degree two over X. The passages from one of these points of view to another are carefully settled in their greatest generality: we only need that 2 is invertible on X. We then restrict to the projective space P_n, and w…
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 …
Let $φ:Σ_1\longrightarrow \mathbb{P}^2$ be a blow up at a point on $\mathbb{P}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\geq 4$ by $φ$, and let $P$ be a point on $C$. Let $π:\tilde{C}\longrightarrow C$ be a double covering branched along the reduced divisor on $C$ obtained as the intersection of $C$ and a reduced divisor in $|-2K_{Σ_1}|$ containing $P$. In this paper, we investigate the Weierstrass semigroup $H(\tilde{P})$ at the ramification point $\tilde{P}$ of $π$ over $P$, in the case where the intersection multiplicity at $φ(P)$ of $φ(C)$ and the tangent lin…
Rank $2$ aCM and Ulrich bundles on Fano and Calabi--Yau double coverings of $\mathbb{P}^3$
本文证明了一类特殊有限覆盖上 aCM 和 Ulrich 层的存在性，精确描述了双重覆盖上秩 $2$ 情况的零轨迹，并在正则情况下完成分类，同时验证了沿度为 $4,6,8$ 除子分岔的 $\mathbb{P}^3$ 双重覆盖上这些层的存在性及稳定时模空间分支维数。
In this paper we prove that the Hurwitz space H_{9,8}, which parameterizes 8-sheeted covers of P^1 by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.
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.
Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $¶^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to study the rank two vector bundles according to the branch curve of the covering which they come from. Thus, in the first part we prove that, given a double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$, the jumping lines of fixed order (order depending on $r$ and on the parity of the rank two vector bundle) of the direct images vector …
An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.
We prove that a double covering of P^3 branched along a very general sextic surface is not stably rational.
Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4χ(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch locus of the covering $S\rightarrow S/i,$ where $i$ is the hyperelliptic involution of $S.$ For $K_S^2<3χ(\mathcal O_S)-6,$ we show how to determine the possibilities for this branch curve. As an application, given $g>4$ and $K_S^2-3χ(\mathcal O_S)<-6,$ we compute the maximum value for $χ(\mathcal O_S).$ This list of possibilities is sharp.
Rational elliptic surfaces with six singular double fibres
本文研究具有六个奇异双纤维的有理椭圆面，通过多种视角进行分类并给出构造方法。
In this note, we introduce the notion of an unramified strongly cyclic covering for a cyclic curve, a class that has similar properties to, and contains, unramified double covers of hyperelliptic curves. We determine several of their basic properties, extending the theorems in \cite{MR2217998} to this larger class. In particular, we will write down equations for smooth affine models, determine when they are isomorphic, and discuss the curves that they are ramified cyclic covers of.
Here we investigate the canonical Gaussian map for higher multiple coverings of curves, the case of double coverings being completely understood thanks to previous work by Duflot. In particular, we prove that every smooth curve can be covered with degree not too high by a smooth curve having arbitrarily large genus and surjective canonical Gaussian map. As a consequence, we recover an asymptotic version of a recent result by Ciliberto and Lopez and we address the Arbarello subvariety in the moduli space of curves.
We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.
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.
The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As …
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian.
Families of cyclic curve coverings with maximal monodromy
本文研究了具有最大单值性的循环曲线覆盖族，通过Hodge结构变分的G分解条件，证明了单值性最大性的判据，并探讨了双伽罗华覆盖和三伽罗华覆盖的测地性问题。
In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which "numerical evidence of modularity" was found in the second Author's thesis.…
Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings
本文利用Levine的二次Riemann-Hurwitz公式，计算几何循环分支覆盖的二次欧拉特征，并在特定情况下将其与基底方案的特征联系起来。
Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of Ulrich line bundles on $X$ with respect to this double covering. Also, for every $s\geq 1$, we describe the classes of such surfaces which admit Ulrich line bundles and give examples.
Explicit bounds on common projective torsion points of elliptic curves
本文在给定素数处约化类型的温和假设下，基于Raynaud对Manin-Mumford猜想的工作并结合对数代数几何技术，给出了椭圆曲线公共射影挠点数量的显式有效界。
Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. The double covering $X \to Y$ is singular in a finite number of points (corresponding to the points $y \in Y$ such that the quadric $Q_y$ degenerates to a union of two planes), the fibers of $M$ over such points are unions of two planes intersecting in a point. The main result of the paper is a cons…
Ulrich bundles on double coverings of projective space
本文利用矩阵分解证明特定有限覆盖存在 Ulrich 层，并通过 Hartshorne--Serre 对应刻画了射影空间二重覆盖上秩 2 Ulrich 层的存在条件、零轨迹及模空间性质。
A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.
In this paper, we classify all the K3 surfaces covering a Kummer surface. Our classification is expressed in terms of period lattices and extends Morrison's criterion of K3 surfaces with a Shioda-Inose structure. Moreover, we list all the K3 surfaces covering a general Kummer surface and provide very geometrical examples of such a phenomenon.
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…
Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have. Caporaso, Harris, and Mazur [CHM97] have shown that the weak Bombieri-Lang conjecture implies that for any number field $F$ and any integer $g \ge 2$ there is an absolute upper bound $B(F; g)$ on the number of points on a genus $g$ curve over $F$. Furthermore, the strong Bombieri-Lang conjecture implies that for each genus $g \ge 2$, there is an absolute bo…
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP^3 branched along singular quartic surfaces. On these threefolds we find families of real lines in explicit form and prove that which families have to be chosen as twistor lines depend on how we take small resolutions of the double covering. This is a first step for determining t…
We extend results on Weierstrass semigroups at ramified points of double covering of curves to any numerical semigroup whose genus is large enough. As an application we strengthen the properties concerning Weierstrass weights in \cited{To}.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。