Xiaoguang Ma, Jian Zhou
We propose a new definition of the elliptic genera for complete intersections, not necessarily nonsingular, in projective spaces. We also prove they coincide with the expressions obtained from Landau-Ginzburg model by an elementary argument.
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We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
Elliptic genera of complete intersection Calabi--Yau 17-folds in $F_4$-Grassmannians
本文在$F_4$-格拉斯曼流形中构造了一对非双有理的Calabi-Yau完全交17维流形，它们具有不同的陈数但相同的椭圆亏格，从而为椭圆亏格的导出不变性问题提供了反例。
A Sage package for calculation of elliptic genera of homogeneous spaces and their complete intersections
本文介绍一个SageMath包，用于计算齐性空间及其完全交的椭圆亏格，涵盖弱Jacobi形式基、Chern数及椭圆亏格的计算，并附有示例，服务于arXiv:2306.10650。
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve
本文通过改进Oberdieck-Pixton的工作，重新证明了Faber在模空间tautological环中关于socle交数的公式，并揭示了新的tautological关系。
New unlikely intersections on elliptic surfaces
本文将复数域上关于 Jacobi 椭圆曲面不可期交集的相切结果扩展至正特征 $p$，并提出了一种研究代数曲线相切性质的新方法，发现了与 Manin 同态相关的新交集族。
In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.
In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.
Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.
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。
Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the Pink-Zilber Conjeture and the Mordel-Lang plus Bogomolov Theorem for curves.
Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zari…
Let $E_λ$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-λ)$. We recently proved that, given $n$ linearly independent points $P_1(λ), \dots,P_n(λ)$ on $E_λ$ with coordinates in $\bar{\mathbb{Q}(λ)}$, there are at most finitely many complex numbers $λ_0$ such that the points $P_1(λ_0), \dots,P_n(λ_0)$ satisfy two independent relations on $E_{λ_0}$. In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two non-isogenous families of elliptic curves and in a family of split semi-abelian v…
In this paper, we study tame Galois coverings of semistable models that arise from torsion points on elliptic curves. These coverings induce Galois morphisms of intersection graphs and we express the decomposition groups of the edges in terms of the reduction type of the elliptic curve. To that end, we first define the reduction type of an elliptic curve $E/K(C)$ on a subgraph of the intersection graph $Σ(\mathcal{C})$ of a strongly semistable model $\mathcal{C}$. In particular, we define the notions of good and multiplicative reduction on subgraphs of the intersection graph of $Σ(\mathcal{C})…
Elliptic genera of level $N$ for complete intersections
本文研究完全交的$N$级椭圆亏格，将其表示为广义二项式系数之和，并针对$c_1$的正负零情况讨论取值，由此导出Todd亏格、$\u0054hat{A}$-亏格和$A_k$-亏格。
By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double $J$-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants.
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a par…
Unlikely and just likely intersections for high dimensional families of elliptic curves
该论文肯定回答了Shou-Wu Zhang的AIM小组提出的问题：模曲线高维乘积中，簇V上为另一簇W的Hecke平移的点在V中稠密，并探讨了维数假设的必要性及有限域上的例外情形。
An arithmetic intersection for squares of elliptic curves with complex multiplication
本研究针对雅可比簇同构于复乘椭圆曲线平方的亏格2曲线，证明其稳定模型坏约化素数满足$p \leq d/4$，给出基于精细化Humbert不变量的素数计算方法，结合已有公式算出素数在稳定模型判别式中的指数，并完成d<100的显式计算与验证。
The construction introduced by Gross, Hacking and Keel allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a…
Elliptic curves in game theory
本文研究了$2\times2$正则型博弈中完全混合依赖均衡的代数几何模型Spohn曲线，证明了其通常为椭圆曲线并给出了可约性分类，基于其$j$-不变量提出了博弈等价的新定义。
Given two elliptic curves, each of which is associated with a projection map that identifies opposite elements with respect to the natural group structure, we investigate how their corresponding projective images of torsion points intersect.
The WDVV equation is satisfied by the genus 0 correlation functions of any topological field theory in two dimensions coupled to topological gravity, and may be used to determine the genus 0 (rational) Gromov-Witten invariants of many projective varieties (as was done for projective spaces by Kontsevich). In this paper, we present an equation of a similar universal nature for genus 1 (elliptic) Gromov-Witten invariants -- however, it is much more complicated than the WDVV equation, and its geometric significance is unclear to us. (Our prove is rather indirect.) Nevertheless, we show that this …
We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when $X$ is a very general complete intersection and $Π_{i=1}^s d_i > 2$ and we find conditions on $n$, $\underline{d}$ and $k$ under which $F_k(X)$ does not contain either rational or elliptic curves. At the end of the paper, we study the case $Π_{i=1}^s d_i = …
On an elliptic surface or threefold, Catalan numbers appear when one tries to compute the autoequivalence group action on the Bridgeland stability manifold. We explain why this happens by identifying a class of equations in the Chow ring of a fibration, where the solutions always involve Catalan numbers.
We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifold…
We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of distinct manifolds. These include both novel elliptic and K3-fibrations and involve Hirzebruch surfaces and their higher dimensional analogues. En route, we generalize the standard techniques of cohomology computation to these generalized complete intersection Calabi-Yau varieties.
We study the geometry of elliptic fibrations given by Weierstrass models resulting from Step 6 of Tate's algorithm. Such elliptic fibrations have a discriminant locus containing an irreducible component $S$, over which the generic fiber is of Kodaira type I$^*_0$. In string geometry, these geometries are used to geometrically engineer G$_2$, Spin($7$), and Spin($8$) gauge theories. We give sufficient conditions for the existence of crepant resolutions. When they exist, we give a complete description of all crepant resolutions and show explicitly how the network of flops matches the Coulomb bra…
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quo…
We present an exhaustive, constructive, classification of the Calabi-Yau four-folds which can be described as complete intersections in products of projective spaces. A comprehensive list of 921,497 configuration matrices which represent all topologically distinct types of complete intersection Calabi-Yau four-folds is provided and can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html . The manifolds have non-negative Euler characteristics in the range 0 - 2610. This data set will be of use in a wide range of physical and mathematical applications. Ne…
On the Linear AFL: The Non-Basic Case
本文通过连通-平展序列，将线性算术基本引理从基本情形推广至所有轨道和同源类，建立了非基本情形到基本情形的归约，并指出其在整体环境中的潜在应用。
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