Co-fibered products of algebraic curves

Fibered varieties over curves with low slope and sharp bounds in dimension three

In this paper, we first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [BS14b]. Led by their conjecture, we focus on finding the lowest possible slope when $n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower bound is $2$. We also obtain a Cornalba-Harris-Xiao type slope inequality for families of surfaces o…

Hodge-Arakelov inequalities for family of surfaces fibered by curves

The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variat…

$\mathbb{A}^1$-cylinders over smooth $\mathbb{A}^1$-fibered surfaces

We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.

Kodaira 维数为一的椭圆面模空间

Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

本文通过墙交叉技术和扭曲稳定映射技术，研究Kodaira 维数为一的椭圆面模空间，构造无限序列的不可约分量并描述其边界及稳定约简步骤。

Varieties fibered by good minimal models

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

曲线上的某些曲面的Hitchin态射

The Hitchin morphism for certain surfaces fibered over a curve

本文证明Chen-Ngô猜想对直纹曲面及非各向同性椭圆纤维化（的爆破）上的任何约化群成立，并进一步对经典群证明了限制在半调和Higgs丛模空间上的Hitchin态射的满射性。

Isolated rational curves on K3-fibered Calabi-Yau threefolds

We study each of the 16 types of complete intersection Calabi-Yau threefolds in projective n-space times the projective line, for various n, and prove existence of isolated rational curves of bidegree (d,0) for all positive integers d on a general threefold of each of these types. As a corollary we find isolated rational curves in all degrees on a general CICY of each of the 5 possible types in usual projective n-space.

Complete moduli for fibered surfaces

Families of stable curves of genus $γ$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_γ$. It is natural to compactify the corresponding moduli problem using stable maps into the stack. In order to get a complete moduli problem, the source curves must acquire extra structure, and you will have to read the paper to find what that is. In this paper, we define these stable maps into $\bar{\cal M}_γ$ in characteristic 0, and show that they form a proper Deligne-Mumford stack admitting a projective coarse moduli space. A comparison with Al…

Chern Classes of Fibered Products of Surfaces

In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For f, a generic projection to CP^2, of an algebraic surface X, we define X_k (for k smaller than degf) to be the k products of X over f minus the big diagonal. For k=degf, X_k is called the Galois cover of f w.r.t. full symmetric group. Let S be the branch curve of f. We give a formula for c_1^2 and c_2 of X_k, in terms of degf, degS, and the number of cusps, nodes and branch points of S. We apply the formula in 2 examples and add a conjecture concerning the spin structure of fibered produ…

Birational automorphisms of algebraic varieties with a pencil of cubic surfaces

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.

由椭圆曲线乘积纤维化的超凯勒四流形

Hyper-Kähler Fourfolds Fibered by Elliptic Products

本文研究射影超凯勒四流形中以两条椭圆曲线乘积为纤维的纤维化，在温和一般性条件下补充了Markushevich对雅可比簇情形的分类。

Sections of Calabi-Yau threefolds with K3 fibration

We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.

CM relations in fibered powers of elliptic families

Let $E_λ$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots , P_n \in E_λ\left(\overline{\mathbb{Q}(λ)}\right)$ we prove that there are at most finitely many complex numbers $λ_0$ such that $E_{λ_0} $ has complex multiplication and $P_1(λ_0), \dots ,P_n(λ_0)$ are dependent over $End(E_{λ_0})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}…

Fujita decomposition and Massey product for fibered varieties

Let $f\colon X\to B$ be a semistable fibration where $X$ is a smooth variety of dimension $n\geq 2$ and $B$ is a smooth curve. We give the structure theorem for the local system of the relative $1$-forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*ω_{X/B}$. We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally we give a criterion to have that $X$ is not …

Fibered Threefolds and Lang-Vojta's Conjecture over Function Fields

Using the techniques introduced by Corvaja and Zannier we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler Characteristic of the base curve.

T-齐次辫子的闭包是实代数的

Closures of T-homogeneous braids are real algebraic

本研究针对实代数链环的纤维化逆猜想，证明了一大类纤维化链环（含T-齐次辫子闭包等）是实代数的，构造了对应半全纯实多项式映射并给出次数信息。

通过嵌入扩张纤维范畴上的幺半群结构

Extending monoidal structures on fibered categories via embeddings

该研究针对小范畴及其满子范畴的嵌入关系，证明了满足自然条件的纤维范畴幺半群结构由子范畴上的限制唯一决定且可扩张，该结论可应用于反常层与可构造导出范畴，也适用于纤维范畴态射及其幺半群性。

Height inequality of algebraic points on curves over functional fields

The purpose of this paper is to give a linear and effective height inequality for algebraic points on curves over functional fields. Our height inequality can be viewed as the logarithmic canonical class inequality of a punctured curve over a functional field (a fibered surface minus a section).

Fundamental gerbes

For a class of affine algebraic groups $\mathcal C$ over a field, we define the notions of $\mathcal C$-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on $\mathcal C$ implying that a fibered category $X$ over $κ$ satisfying mild hypotheses admits a Nori $\mathcal C$-fundamental gerbe. We show that these are verified in particular by the classes of virtually abelian and virtually unipotent group schemes. In the second situation, under a properness condition on $X$, we give a tannakian interpretati…

通过因子分解构造纤维化范畴上的幺半群结构

Constructing monoidal structures on fibered categories via factorizations

本研究针对小范畴上纤维化范畴的幺半群结构构造问题，提出利用光滑态射与闭浸入对应的子范畴限制的因子分解方法，证明满足相干条件的限制对可唯一确定整体幺半群结构，并给出适配反常层的逆像替换直像的变体方法。

Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces

We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Finally we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P)^3$.

Hyperbolic Secant Varieties of M-Curves

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. Fo…

Slope equality of plane curve fibrations and its application to Durfee's conjecture

We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.

射影直线阿贝尔覆盖的Prym轨迹中Shimura曲线的存在性

On the Existence of Shimura curves in the Prym locus of abelian covers of projective line

该研究借助希格斯丛理论与Viehweg-Zuo刻画，证明了g≥8时射影直线全分歧Z_{2p}等类型阿贝尔覆盖的Prym轨迹中不存在非紧Shimura曲线，p为不小于5的素数。

Families of algebraic varieties and towers of algebraic curves over finite fields

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.

代数簇的合冲与代数曲线的对称积

Syzygies of algebraic varieties through symmetric products of algebraic curves

本文综述了利用代数曲线对称积的几何方法证明的关于代数簇合冲的若干近期结果，包括亏格猜想、割线簇与切发展曲面的合冲、格林猜想及渐近合冲。

Topology of plane algebraic curves: the algebraic approach

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

纤维化范畴上的Drinfeld-Lau下降

Drinfeld-Lau Descent over Fibered Categories

本文将Drinfeld引理推广至真代数栈、仿射芽及任意纤维化范畴，证明了在多种广义栈条件下自然函子均为等价。

An Algebraic Analysis of Conchoids to Algebraic Curves

We study conchoids to algebraic curve from the perspective of algebraic geometry, analyzing their main algebraic properties. We introduce the formal definition of conchoid of an algebraic curve by means of incidence diagrams. We prove that, with the exception of a circle centered at the focus and taking $d$ as its radius, the conchoid is an algebraic curve having at most two irreducible components. In addition, we introduce the notions of special and simple components of a conchoid. Moreover we state that, with the exception of lines passing through the focus, the conchoid always has at least …

具有两个Galois点的代数曲线的后代

Descendants of algebraic curves admitting two Galois points

本文研究代数曲线的Galois点与商曲线的联系，引入具有两个Galois点的后代概念，并证明Fermat曲线的后代均为Fermat曲线，其次数为素数时无后代。

Real algebraic curves, the moment map and amoebas

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications.

Clifford Theorem for real algebraic curves

We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

A Compactness Criterion for Real Plane Algebraic Curves

Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.

A Theory of Duality for Algebraic Curves

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of duality, we prove generalisations of Plucker's formulae for algebraic curves. The results hold for arbitrary characteristic of the base field L, with some occasional exceptions when characteristic(L)=2, which we observe in the course of the paper.

在OSCAR中绘制实平面代数曲线

Drawing real plane algebraic curves in OSCAR

本文展示如何利用OSCAR计算机代数系统生成拓扑正确或视觉美观的实平面代数曲线图示。

A note about invariants of algebraic curves

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.

Lipschitz geometry of complex plane algebraic curves

We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.

The join of algebraic curves

Let X,Y be algebraic curves in P^n over C. We give an effective description of the join J(X,Y)\in P^n of X and Y in terms of local parametrizations of X and Y.

A non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence

We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[(-1)*C] in JC modulo algebraic equivalence is torsion.

Algebraic curves with automorphism groups of large prime order

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also give some partial results on $q$-curves for $q = g, g-1$.

On algebraic equivalences among the 27 Abel-Prym curves on a generic abelian 5-fold

This article shows that on a generic principally polarized abelian variety of dimension five the $\mathbb{Q}$-vector space of algebraic equivalences among the 27 Abel-Prym curves has dimension 20.

Tropical and algebraic curves with multiple points

Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.

Generic symmetry defect set of an algebraic curve

Let $X \subset \mathbb{C}^{2n}$ be an $n$-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of $X$. Moreover, we compute its singularities for $X_d$ being a generic curve of degree $d$ in $\mathbb{C}^2$.

代数曲线的利普希茨嵌入

Lipschitz embeddings of algebraic curves

研究实代数曲线利普希茨分类，证明其内部与\(\mathbb{R}^3\)中具LNE连通分量的外部分类等价，找到实仿射代数曲线的LNE模型，还指出复范畴不成立此结论。

Injective linear series of algebraic curves on quadrics

We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

On topological invariants of algebraic threefolds with ($\mathbb Q$-factorial) singularities

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the…

On the separate algebraicity along the families of algebraic curves

A new generalization of the classical separate algebraicity theorem is suggested and proved.

Arf invariants of real algebraic curves

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

Euclid meets Bezout: Intersecting algebraic plane curves with the Euclidean algorithm

We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.

Rational transformations of algebraic curves and elimination theory

Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic cu…

8次代数曲线与辛曲线

Algebraic and symplectic curves of degree 8

本文研究8次不可约射影平面曲线在代数与辛两种情形下具有指定奇点拓扑类型的存在性问题。

A Theory of Divisors for Algebraic Curves

The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for Algebraic Curves",(*), where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from the cited paper, (*), in order to develop a refined theory of g_{n}^{r} on an algebraic curve. The refinement depends critically on relacing the notion of a point with a branch. This allows us to construct a theory of diviso…

Vertex Algebras and Algebraic Curves

This is the text of the Bourbaki seminar that I gave on June 24, 2000.

Incidence systems on Cartesian powers of algebraic curves

We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.