On certain families of elliptic curves in projective space

Moduli of elliptic curves in products of projective spaces

We exhibit a smooth compactification of the moduli space of elliptic curves in a product of projective spaces with tangency along a subset of its toric boundary divisors. This is a Vakil--Zinger type of desingularization for maps to a product of projective spaces using ideas of elliptic singularities and logarithmic geometry, extending the recent work by Ranganathan--Santos-Parker--Wise. We use this to construct the virtual fundamental classes of the spaces of genus 1 maps to a special class of simple normal crossings pairs.

The projective characterization of elliptic plane curves which have one place at infinity

In this paper we consider smooth affine elliptic plane curves having one place at infinity. We identify them with elliptic projective plane curves having only one cusp as their singular points and meeting with the line at infinity only at the cusp. We characterize such curves by the self-intersection number of the strict transform of them via the minimal embedded resolution of their cusp. Furthermore, we prove that the self-intersection number of them is the maximum value among those of all the elliptic plane curves having only one cusp.

Hodge sheaves underlying flat projective families

We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …

Arithmetic of a fake projective plane and related elliptic surfaces

The purpose of the present paper is to explain the fake projective plane constructed by J.H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J.H. Keum as well as the one constructed by M.N. Ishida in terms of p-adic uniformization.

On the isotriviality of families of elliptic surfaces

We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kovács, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references.

Unirationality of varieties described by families of projective hypersurfaces

Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect to $d$, $r$ and $t$, then $\mathscr{X}$ is unirational. This extends results in \cite {Pre, HMP}.

Reducible Families of Curves with Ordinary Multiple Points on Surfaces in Projective Three-Space

In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves on surfaces in projective three-space with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness i…

Geometry of obstructed equisingular families of projective hypersurfaces

We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the…

Complete families of immersed curves

We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space, completing the picture in projective space. In another direction, we isolate the necessary positivity condition on the tangent bundle of projective space used to run the argument, which allows us to rule out large dimensional families of immersed curves of positive genus in generalized flag varieties.

On abelian families and holomorphic normal projective connections

We prove Chern class equalities for abelian families with a holomorphic normal projective connection.

New families of rank 2 bundles on projective space

We present a new family of monads whose cohomology is a stable rank two vector bundle on $\PP$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. Such facts are used to prove that the moduli space of stable rank two vector bundles of zero first Chern class and second Chern class equal to 5 has exactly three irreducible components.

On the isotriviality of families of projective manifolds over curves

Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-a…

Irregular elliptic surfaces of degree 12 in the projective fourspace

So far only a few families of smooth irregular surfaces are known to exist in P^4 up to pullbacks by suitable finite morphisms from P^4 onto P^4 itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in P^4. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case.

Elliptic Curves with Large Intersection of Projective Torsion Points

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

投影流形的变形刚性和光滑族的等变性

Deformation rigidity for projective manifolds and isotriviality of smooth families

研究投影流形的变形刚性及光滑族的等变性，证明在特定条件下所有纤维均同构于固定投影流形，并建立新的Parshin-Arakelov型等变性准则。

椭圆曲线覆盖的完整通用族

Complete Families of Generic Covers of Elliptic Curves

该研究提出一个关于椭圆曲线覆盖完整族的猜想，并探讨其对模空间斜率下界的影响。

Splitting submanifolds of families of fake elliptic curves

Let N a compact complex submanifold of a compact complex manifold M. We say N splits in M, if the holomorphic tangent bundle sequence splits holomorphically. By a result of Mok a splitting submanifold of a Kaehler Einstein manifold with a projective structure is totally geodesic. The classification of all splitting submanifolds of families of fake elliptic curves given here completes the case of threefolds M with a projective structure by a previous result of the authors.

Invariance of Gamma-dimension for projective families

This article is withdrawn because of a mistake in the main result of the paper.

Families of genus two curves with many elliptic subcovers

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which…

椭圆曲线公共射影挠点的显式界

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。

On covering and quasi-unsplit families of rational curves

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.

通过四维射影Fano超曲面的椭圆Gromov-Witten不变量枚举椭圆曲线

Enumeration of Elliptic Curves via Elliptic Gromov-Witten Invariants of Four Dimensional Projective Fano Hypersurfaces

本文通过四维射影Fano超曲面的椭圆GW不变量，提出一个猜想，阐明椭圆曲线数量与GW不变量之间的关系。

某些有理椭圆射影四维簇的拓扑刻画与 Hodge 结构

Topological characterization and Hodge structures of some rationally elliptic projective fourfolds

研究嵌入 $\mathbb{P}^8$ 的有理椭圆射影四维簇，借助法丛 Chern 数等刻画其拓扑并证明同构于 $\mathbb P^4$ 的条件，同时验证 Hodge 猜想及非正 Hodge 层级。

On the families of hyperplane sections of some smooth projective varieties

In this note, we give two applications of \cite[Theorem 3.1]{Hwang}. We first study the free family $\mathcal{K}$ of hyperplane sections of the smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d\ge 3$. We prove that $X$ is determined by the free family $\mathcal{K}$ if $\dim(X)\ge 4$. As an application, we deduce that for $n\ge 4$, the hyperplane section of $X$ varies maximally in the moduli space of the smooth hypersurface of degree $d\ge 3$ in $\mathbb{P}^n$. We then study the free family of hyperplane sections of the smooth projective surface $X$ with Kodaira dimension $κ(X)\ge 0$.…

Constancy of Newton polygons of $F$-isocrystals on Abelian varieties and isotriviality of families of curves

We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of projective smooth families of curves over Abelian varieties. We also study the isotriviality over simply connected projective smooth varieties.

On Families of Rational Elliptic Surfaces with J-Invariant Functions of Degree One

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…

Families of elliptic curves with genus 2 covers of degree 2

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic different from 2 is birational to a product of two elliptic curves over the projective line. We formulate and prove a generalization of this theorem for the relative si…

Equisingular Families of Projective Curves

In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all projective curves with prescribed singularities in a fixed linear system. We also discuss the analogous problem for hypersurfaces of arbitrary dimension with isolated singularities, and we close with a section on open problems and conjectures.

Monodromy of elliptic surfaces

We study monodromy groups of elliptic fibrations over the projective line.

Projective varieties covered by isotrivial families

Let X,Y be projective schemes over a discrete valuation ring R, where Y is generically smooth and g:X \to Y a surjective R-morphism such that g_*\mathcal{O}_X = \mathcal{O}_Y. We show that if the family X \to Spec(R) is isotrivial, then the generic fiber of the family Y \to Spec(R) is isotrivial.

Geometric Variations of Local Systems and Elliptic Surfaces

Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of Kähler manifolds for which each fibre itself is fibred by codimension one Kähler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livné in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the univers…

Stable rationality in smooth families of threefolds

We exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.

射影超曲面族的退化与霍奇猜想

Degeneration of families of projective hypersurfaces and Hodge conjecture

该研究通过维数归纳法，结合光滑超曲面情形的既有结论、混合霍奇模的邻近循环函子等工具，证明了光滑复射影簇的霍奇猜想。

Families of elliptic curves in $\mathbb{P}^3$ and Bridgeland Stability

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blow up with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

Feigin–Odesskii 与 Polishchuk 所定义的射影空间上椭圆 Poisson 括号的辛叶

The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

本文研究 Feigin–Odesskii–Polishchuk 在复射影空间上定义的椭圆 Poisson 结构，并利用椭圆正规曲线的高次割线簇刻画其全部辛叶。

Families of elliptic curves with rational torsion points of even order

This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.

允许耦合凯勒-爱因斯坦度量但不允许可爱因斯坦度量的投影包

Families of projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics

本文构造了允许耦合凯勒-爱因斯坦度量但不允许可爱因斯坦度量的高维托里克粉丝 manifold，并提出在所有维度n≥4中此类例子存在的猜想。

Projective bundles of singular plane cubics

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in a previous paper that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generall…

Nonabelian elliptic Poisson structures on projective spaces

We review nonabelian Poisson structures on affine and projective spaces over $\mathbb{C}$. We also construct a class of examples of nonabelian Poisson structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson structures depend on a modular parameter $τ\in\mathbb{C}$ and an additional descrete parameter $k\in\mathbb{Z}$, where $1\leq k<n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(τ)$.

椭圆虚拟结构常数与完备交集在加权射影空间中的格罗莫夫-王特征值

Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space

本文研究了加权射影空间中完备交集的椭圆虚拟结构常数及其与格罗莫夫-王特征值的关系。

On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties

For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-Păun. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbol…

复椭圆曲面在射影直线上同调格与周期的一种半数值算法

A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line

本文提出一种半数值算法，用于计算复射影直线上椭圆曲面的同调有效基并进行周期积分，从而启发式恢复Néron-Severi格、超越格及Mordell-Weil群与格等代数不变量，并提供SageMath实现。

A graphical method to calculate Selmer groups of several families of non-CM elliptic curves

In this paper, we extend the ideas of Feng [F1], Feng-Xiong [FX] and Faulkner-James [FJ] to calculate the Selmer groups of elliptic curves $ y^{2} = x (x + \varepsilon p D) (x + εq D). $

On several families of elliptic curves with arbitrary large Selmer groups

In this paper, we calculate the $ ϕ(\hatϕ)-$Selmer groups $ S^{(ϕ)} (E / \Q) $ and $ S^{(\hatφ)} (E^{\prime} / \Q) $ of elliptic curves $ y^{2} = x (x + εp D) (x + εq D) $ via descent theory (see [S, Chapter X]), in particular, we obtain that the Selmer groups of several families of such elliptic curves can be arbitrary large.

控制椭圆曲线的射影簇中的非阿基米德整曲线

Non-Archimedean entire curves in projective varieties dominating an elliptic curve

本文针对允许支配态射到椭圆曲线的无群射影曲面，证明其 K-解析 Brody 双曲性，从而建立不规则度为一情形的非阿基米德 Green–Griffiths–Lang 猜想。

Canonical lifts of families of elliptic curves

We show that the canonical-lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti an…

Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation

In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

Mordell-Weil ranks of families of elliptic curves parametrized by binary quadratic forms

We prove results on the Mordell--Weil rank of elliptic curves $y^2=x(x-αa^2)(x-βb^2)$ parametrized by binary quadratic forms $αa^2+βb^2=γc^2$. We express our explicit lower bounds over number fields and offer a detailed description of the corresponding Mordell-Weil group structure in the function field case.

Families of Calabi-Yau elliptic fibrations in $\mathbb{P}(\mathcal{L}^a \oplus \mathcal{L}^b \oplus \mathcal{O}_B)$

Let $B$ be a smooth projective surface, and $\mathcal{L}$ an ample line bundle on $B$. The aim of this parer is to study the families of elliptic Calabi--Yau threefolds sitting in the bundle $\mathbb{P}(\mathcal{L}^a \oplus \mathcal{L}^b \oplus \mathcal{O}_B)$ as anticanonical divisors. We will show that the number of such families is finite.

通过p进霍奇理论与朗兰兹对应构造4个穿孔的复射影直线上的$\text{GL}_2$型阿贝尔簇族及其在Painleve VI方程代数解中的应用

Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation

本研究借助p进霍奇理论与朗兰兹对应，构造了4穿孔射影直线上无穷多族带特定坏约化的$\text{GL}_2$型阿贝尔簇，结合相关猜想证明其可由椭圆曲线泛族挠截面参数化，且可导出Painleve VI方程的代数解。

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. V.2: Main theorem is corrected for small finite fields.

关于Kähler族中伴随类的伪有效性与体积的研究

On Pseudo-Effectivity and Volumes of Adjoint Classes in Kähler Families with Projective Central Fiber

本文研究了Kähler族中伴随类的伪有效性与体积的变形行为，证明了体积的局部不变性并验证了Siu的不变性猜想。

具有理性椭圆空间性质的奇异代数簇族

Families of singular algebraic varieties that are rationally elliptic spaces

本文研究了具有有限理性同调秩和同调群秩之和的奇异代数簇族，证明了此类无限光滑理性椭圆 3 折叠族不存在。

Toward a geometric construction of fake projective planes

We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a $(2,3)$-elliptic surface that is the minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of ${\mathbb Q}$-homology projective planes with cusps only.