Gromov elliptic resolutions of quartic double solid
Ciro Ciliberto, Mikhail Zaidenberg
本文构造节点四次二重立体实例，证明其存在一致有理且在 Gromov 意义下为椭圆的小型代数消解。
我们构造了若干节点四次二重立体的例子，它们容许一致有理、因而在 Gromov 意义下为椭圆的小型代数消解。
We construct examples of nodal quartic double solids that admit uniformly rational, and so elliptic in Gromov' sense, small algebraic resolutions.
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Algebraic ellipticity of quartic double spaces
本文证明特征零代数闭域上所有光滑四次双n重（n≥2）在格罗莫夫意义下代数椭圆，特别回答四次双立体情形的问题4.28，并证明移除余维数至少二的闭子集保持该性质。
The Gromov-Witten and Donaldson-Thomas correspondence for trivial elliptic fibrations
本文研究了文献[MNOP 1，MNOP 2]中关于平凡椭圆纤维化的Gromov-Witten和Donaldson-Thomas对应.特别地，我们证明了当三重为$E \times S$时，准素场的Gromov-Witten和Donaldson-Thomas对应，其中$E$是光滑椭圆曲线，$S$是具有数值平凡正则类的光滑曲面.
Algebraic Gromov ellipticity: a brief survey
本文简短综述了 Gromov 意义下的代数椭圆簇研究。
Gromov's ellipticity of principal $G_m$-bundles
证明完备一致有理簇上的每个非平凡主$G_m$-丛在格罗莫夫意义下具有代数椭圆性，解决了相关代数几何问题。
Algebraic Gromov's ellipticity of cubic hypersurfaces
本文证明了 P^{n+1}（n>1）中的光滑三次超曲面在 Gromov 意义下具有代数椭圆性，首次给出了此类非有理射影流形的实例，并推导出其穿孔仿射锥也具有椭圆性。
We find classes of projective manifolds that are elliptic in the sense of Gromov and such that the affine cones over these manifolds also are elliptic off their vertices. For example, the latter holds for any generalized flag manifold of dimension at least 3 successively blown up in a finite set of points and infinitesimally near points. This also holds for any smooth projective rational surface. For the affine cones, the Gromov ellipticity is a much weaker property than the flexibility. Nonetheless, it still implies the infinite transitivity of the action of the endomorphism monoid on these c…
We show that a general elliptic K3 surface with a section is determined uniquely by its discriminant, which is a configuration of 24 points on the projective line. It follows that a general elliptic K3 surface with a section can be reconstructed from its Gromov-Hausdorff limit as the volume of the fiber goes to zero.
Enumeration of surfaces containing an elliptic quartic curve
在三维射影空间中，一个非常一般的至少为四次的曲面除了与曲面相交外不包含任何曲线。我们找到了一个公式的次数的轨迹曲面的次数至少为五次，其中包含一些椭圆四次曲线。我们还计算的程度的轨迹的四次曲面包含一个椭圆四次曲线，一种情况下不包括该公式。
Quartic equations and 2-division on elliptic curves
设K是一个不同于2的特征域，C是K上由Weierstrass方程给出的椭圆曲线。要将群C中的一个元素除以2，必须解一个四次方程。我们将从这个过程中产生的四次曲线，并找出四次曲线在多大程度上决定了曲线和点。我们发现来自2-分裂的2-和3-扭点的四次曲线，并将这种对应关系推广到奇异平面三次曲线。我们使用这些结果来研究的问题，其中度4地图的曲线可以实现作为重复的多节椭圆曲面上。
Ramified and Split models of elliptic surfaces and bitangent lines of quartic curves
本文定义并研究了椭圆曲面的分歧模型与分裂模型，重点关注某些有理椭圆曲面，并借此对至多只有节点的不可约四次曲线的双切线给出新的观察。
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。
We study the quasimap invariants of elliptic and K3 fibrations. Oberdieck and Pixton conjectured that the Gromov-Witten potentials of elliptic fibrations are quasi-modular forms. Analogously, we propose similar conjecture for the quasimap potentials of elliptic fibrations. We also conjecture some finite generation properties of quasimap potentials of K3 fibrations. Via wall-crossing conjecture, this will imply some quasi-modularity of the Gromov-Witten potentials of K3 fibrations. We provide some evidences for our conjectures through several examples. The method here can be further generalized…
Elliptic Gromov - Witten invariants and the generalized mirror conjecture
给出了半单Frobenius结构和复振荡积分的亏格为1的Gromov-Witten不变量的镜像表示猜想。证明了具有孤立不动点的紧致Kähler流形上的环面等变Gromov-Witten不变量和这类流形上的凹丛空间上的环面等变Gromov-Witten不变量.关于亏格0的Gromov-Witten理论的几个结果包括：一个非线性Serre对偶定理，它在亏格0镜像猜想中的应用，环面流形上凹丛空间的镜像定理，推广了B. Lian，K. Liu和S.- T.你。我们还建立了一个对应（见第4节中的广泛脚注）之间的新证明的亏格0镜像猜想的五次3倍和我们的证明相同的猜想两年前。
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。
Jacobian elliptic fibrations on the generalized Inose quartic of Picard rank sixteen
本文研究 Picard 格包含 $H \oplus E_7(-1) \oplus E_7(-1)$ 的复代数 K3 曲面族，证明其一般成员恰有四个互不等价的 Jacobian 椭圆纤维化，并构造了显式线束。
We obtain a formula for the number of genus one curves with a variable complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done using Getzler's relationship among cohomology classes of certain codimension 2 cycles in $\overline{M}_{1,4}$ and recursively computing the genus-one Gromov-Witten invariants of del Pezzo surfaces. Using completely different methods, this problem has been solved earlier by Bertram and Abramovich, Ravi Vakil, Dubrovin and Zhang and more recently using Tropical geometric methods by …
The elliptic Gromov-Witten invariants of CP^3
我们给出了两个明确的递归式，它们用有理不变量来确定CP ^3的椭圆Gromov-Witten不变量，并给出了一个直到5次的表。与有理Gromov-Witten不变量不同，系数是负数和分数。本文进一步证明了N^1_ab +（2n-1）N^0_ab/12是通过a条一般直线和b条一般点的椭圆曲线的个数。
Degenerations of elliptic quartics and K-moduli of Fano threefolds
本文研究经两次爆破构造的法诺三维簇的K-模空间，将其同构于带CM线丛线性化的VGIT商空间，分类了该形变族的K-(半/多)稳定成员，相关证明用到希尔伯特概型几何等新要素。
In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial invariants. We give an example of wall-crossing phenomenon of the invariant and expect it satisfies Kontsevich-Soibelman wall-crossing formula.
An algorithm to determine all the Gromov-Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro type constraints that allow to determine all the other Gromov-Witten invariants in terms of the stationary ones. In the case of an elliptic curve, we show that these Virasoro type constraints can be explicitly solved leading to a very explicit formula for the full Gromov-Witten potential in terms of the stationary invariants.
On degree zero elliptic orbifold Gromov-Witten invariants
我们用两种方法计算了亏格为一的具有非堆叠插入的零次轨道Gromov-Witten不变量，这是子方程和因子方程的例外情况。一种方法涉及相关模空间的详细分析。另一种方法是虚拟定位法，它适用于具有孤立不动点的环面作用。这些计算验证了这些不变量的几何评价。还计算了一些亏格为1的扭曲轨道Gromov-Witten不变量。
Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G <= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields.
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。
Torsion of elliptic curves with rational $j$-invariant over quartic number fields
本文分类了定义在四次数域上且具有有理 $j$-不变量的椭圆曲线的挠子群的可能群结构。
Gromov-Witten Generating Series of Elliptic Curves as Configuration Space Integrals
本文为椭圆曲线的Gromov-Witten生成级数提供了新的分区求和公式，证明其为庞加莱丛上同调类的构型空间积分，并探讨了其拟椭圆与拟模性质。
Exceptional elliptic curves over quartic fields
研究了具有给定挠群$T$的固定四次域$K$上直到同构的椭圆曲线的个数.令$T=\Z/m\Z \oplus \Z/n\Z$，其中$m|n$是挠群，使得模曲线$X_1(m,n)$是椭圆曲线。设$K$是一个数域，使得存在正的有限个数的$K$上的椭圆曲线$E_T$，其子群为$K$.我们称这种对为$(E_T, K)$ \emph{exceptional}。已知当$K$在所有二次或三次域上变化时，只有100多个例外对。证明了当$K$在所有四次域上变化时，当$T=\Z/14\Z$或$\Z/15\Z$时存在无穷多个例外对，否则存在无穷多个例外对.
Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs
研究弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何结构，通过理性椭圆面和扎里斯基对构造度数为7和8的扎里斯基对。
Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
本文证明了权为（3，3，3），（4，4，2），（6，3，2）的椭圆轨道线的GW不变量是拟模形式.我们的方法是基于Givental的更高的属重建形式主义适用于设置斋藤的Frobenius结构简单的椭圆奇点。我们的结果是一个更大的项目的一部分，其目标是证明简单的椭圆奇点的朗道-金兹伯格/卡拉比-丘对应。对应关系描述了Gromov-Witten理论（某个超曲面）和Fan-Jarvis-Ruan-Witten理论（某个Landau-Ginzburg势）之间的关系。粗略地说，主要的陈述是，简单椭圆奇点的斋藤的弗罗贝纽斯流形有一些特殊的点，使得在这些点附近的弗罗贝纽斯结构控制着两个理论之一。对应关系的局部部分在M. Krawitz和Y.沈，我们在这里描述的是全球的情况。
Gromov-Witten Generating Series of Elliptic Curves and Iterated Integrals of Eisenstein-Kronecker Forms
本文计算了由Kronecker theta函数构造的Eisenstein-Kronecker形式的各种迭代积分，并将其与椭圆曲线的Gromov-Witten不变量生成级数联系起来。
Algorithms for Gromov-Witten Invariants of Elliptic Curves
提出一种基于代数与tropical几何对应的高效算法，利用Singular和OSCAR计算椭圆曲线Gromov-Witten不变量及Hurwitz数的生成级数，显著提升计算效率并促进其数论与几何性质研究。
The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves
本文提出通过椭圆曲线的Gromov-Witten理论构造量子可积层次，并首次展示包含费米子场的协同场论中的非平凡量子可积层次实例。
We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic…
Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities
我们考虑简单椭圆奇点的镜像对称性。特别是，我们构造同构的Frobenius流形之间的一个从Gromov-Witten理论的加权射影线，一个从理论的原始形式的一个普遍的展开一个简单的椭圆奇点和一个从不变的理论椭圆外尔群。作为结果，我们给出了K所考虑的η积的傅里叶系数的几何解释。齐藤
Elliptic Gromov-Witten Invariants And Virasoro Conjecture
对Eguchi-Hori-Xiong和S. Katz预言Gromov-Witten不变量的生成函数被无穷多个形成Virasoro代数的半分支的微分算子湮灭。本文研究了该猜想的亏格1情形。特别地，我们将给出亏格-1 Virasoro猜想成立的一些充要条件。
We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to $\mathbb{P}^1$) are $E_8 \times E_8$ quasi-bi-Jacobi forms and satisfy a h…
Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten invariants
WDVV方程可以由任何二维拓扑场论的亏格0关联函数满足，并且可以用来确定许多射影簇的亏格0（有理）Gromov-Witten不变量（就像Kontsevich对射影空间所做的那样）。在本文中，我们提出了一个方程的一个类似的普遍性质的亏格1（椭圆）Gromov-Witten不变量-然而，它是更复杂的比WDVV方程，它的几何意义是不清楚我们。(Our证明是间接的）。然而，我们表明，这个方程足以确定椭圆Gromov-Witten不变量的射影空间。在本文的续篇中，我们将证明这个方程是唯一一个不同于WDVV方程的方程，它涉及二维拓扑场论与拓扑引力耦合的椭圆和有理关联函数。目前还不清楚在高等亏格的小相空间上是否存在这类方程，但我们认为这是不可能的。(The亏格0和亏格1的情况是特殊的，因为小相空间上的关联函数确定大相空间上的那些。
Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves
研究四次残差符号海克勒L函数在中心点非消失的比例，以及与椭圆曲线秩的关系。
$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields
本文给出了非阿贝尔三次和四次域上椭圆曲线族的两种构造，在驯顺符号核中分别获得三个和四个线性无关的“积分”元素，证明了不忽略挠元的新积分性判据，并对90余条曲线数值验证了 Beilinson 猜想。
Birational automorphisms of nodal quartic threefolds
众所周知，非奇异极小三次曲面是双有理刚性的;它的双有理自映射群由双正则自映射和双有理对合生成，使得后者之间的所有关系都隐含在椭圆曲线上的反射之间的标准关系中。我们还知道一个阶乘节四次三重函数是双有理刚性的，它的双有理自映射群是由双正则自映射和某些双有理对合生成的。我们证明了这些对合之间的所有关系都隐含在椭圆曲线上的标准关系中，完成了非闭域上的双有理刚性的证明，并描述了一些双有理对合成为正则对合时的情况（特别是，完成了双有理刚性初始定理的证明，因为一些细节在M.Mella的原始论文中没有建立）。
Tate-Shafarevich results for quartic twists in characteristic $2$
本文研究偶特征函数域上的椭圆曲线，通过具4阶自同构的极大曲线的函数域构造$\u211d_2$上超奇异椭圆曲线的四次扭转，给出由相关曲线亏格表示的Mordell-Weil秩公式，以得到任意大秩的椭圆曲线。
A class of geometrically elliptic fibrations by plane projective quartic curves
本文研究特征2下算术亏格3、几何亏格1的非超椭圆曲线纤维化，在特定奇点假设下给出双有理等价的完整分类，证明其可被椭圆纤维化覆盖且覆盖映射具特定双有理与纯不可分性质。
Classification of torsion of elliptic curves over quartic fields
本文确定了四次域上椭圆曲线挠群的所有可能情况，证明不存在零散挠群，并利用秩0方法、Hecke筛法和全局方法等证明了相关模曲线无非尖点四次点。
Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphi…
We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants for those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.
A parametrization of the theta divisor of the quartic double solid
设M（2; 0，3）是Fano三重体X上的秩2稳定向量丛的模空间，其中Chern类c_1=0，c_2=3.本文证明了由X上光滑椭圆五次曲线通过Serre构造得到的向量丛构成M（2; 0，3）的不可约分支M'的开部分，以及到由第二类Chern定义的中间Jacobi J（X）的Abel-Jacobi映射F：M'-->J（X）在θ因子的平移上是84次一般有限的.我们还证明了一般X上的椭圆五次线族是不可约的，维数为10。
We prove that Gromov's ellipticity condition $\mathrm{Ell}_1$ characterizes Oka manifolds. This characterization gives another proof of the fact that subellipticity implies the Oka property, and affirmative answers to Gromov's conjectures. As another application, we establish the localization principle for Oka manifolds, which gives new examples of Oka manifolds. In the appendix, it is also shown that the Oka property is not a bimeromorphic invariant.
Gromov-Witten theory and cycle-valued modular forms
本文证明了权为（3，3，3），（4，4，2），（6，3，2）的椭圆轨道线的Gromov-Witten圈的母函数是圈值拟模形式.这是Milanov和Ruan在圈值水平上工作的推广。首先，我们构造了一个简单椭圆奇点（模扩张问题）的整体上同调域论（CohFT），并证明了它的模性。然后，我们利用Teleman的重构定理证明了圈值水平上的镜像定理，并将其与相应轨道的Gromov-Witten CohFT相匹配，解决了Gromov-Witten CohFT的扩张性质，并导出了其模性.
Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $\mathbb{p}^1$
本文证明了椭圆轨道双折叠$\mathbb{P}^1$的Gromov-Witten不变量的收敛性，并证明了$(3,3,3), (4,4,2)$型和$(6,3,2)$型$\mathbb{P}^1$的Gromov-Witten不变量的收敛性.我们还证明了Gromov-Witten理论对这些轨道折叠的镜像定理和椭圆奇点的FJRW理论。利用T.Milanov和Y.阮的工作，我们证明了上述三种椭圆轨道$\mathbb{P}^1$的所有属的Landau-Ginzburg/Calabi-Yau对应。
We recall that a smooth ample surface $\mathcal{S}$ in a general $(1,2,2)$-polarized abelian threefold, which is the pullback of the Theta divisor of a smooth plane quartic curve $\mathcal{D}$, is a surface isogenous to the product $\mathcal{C} \times \mathcal{C}$, where $\mathcal{C}$ is a genus $9$ curve embedded in $\mathbb{P}^3$ as complete intersection of a smooth quadric and a smooth quartic. We show that the space of global holomorhic sections of the canonical bundle of this surface is generated by certain determinantal bihomogeneous polynomials of bidegree $(2,2)$ on $\mathbb{P}^3$, whi…
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。
Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface
本文研究有理椭圆曲面的Gromov-Witten不变量的母函数（预势）。应用局部镜像原理计算了前势，证明了亏格为0和亏格为1的全纯反常方程。我们提出的全纯异常方程的所有属，并应用它来确定高的亏格Gromov-Witten不变量和表面上的BPS状态。推广了曲面上$g$点的Hilbert格式的Göttsche公式，我们发现我们的结果与Gopakumar和Vafa最近提出的建议（hep-th/9812127）精确一致.
Mock modularity of log Gromov--Witten Invariants: the mirror to $\mathbb{P}^2$
本文研究了对数 Gromov-Witten 不变量的生成函数的模性，证明其为 mock 模形式，并建立了与 $\mathbb{P}^2$ 镜像的对应关系。
We compute the $g=1, n=1$ B-model Gromov-Witten invariant of an elliptic curve E directly from the derived category D(E). More precisely, we carry out the computation of the categorical Gromov-Witten invariant defined by Costello using as target a cyclic $A_\infty$ model of D(E) described by Polishchuk. This is the first non-trivial computation of a positive genus categorical Gromov-Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov-Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
Braid groups, elliptic curves, and resolving the quartic
本文通过分类少股辫群到 PSL2Z 的同态并结合复分析与 Teichmüller 理论，完全分类了配置空间间的全纯映射及椭圆曲线族，并证明了 Castel 关于三股辫群自同态的猜想。
Toroidal 3-orbifolds $(S^1)^6/G$, for $G$ a finite group, were some of the earliest examples of Calabi-Yau 3-orbifolds to be studied in string theory. While much mathematical progress towards the predictions of string theory has been made in the meantime, most of it has dealt with hypersurfaces in toric varieties. As a result, very little is known about curve-counting theories on toroidal orbifolds. In this paper, we initiate a program to study mirror symmetry and the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for toroidal orbifolds. We focus on the simplest example $[E^3/μ_3],$ where $…
Varieties covered by affine spaces, uniformly rational varieties and their cones
本研究将仿射锥的格罗莫夫椭圆性结论，从旗流形、有理光滑射影曲面推广到逐次爆发后的光滑射球星形簇、一致有理射影簇及稳定一致有理完全簇。
Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
Bershadsky-Cecotti-Ooguri-Vafa（BCOV）提出镜像对称的B模型应该用卡-丘簇上的量子场论来描述，他们称之为Kodaira-Spenser理论（我们称之为BCOV理论）。这是我们构建和分析量子BCOV理论的三篇论文中的第一篇。本文构造了任意维Calabi-Yau簇上的经典域论，定义了量子化的含义，分析了Gromov-Witten理论的Givental辛形式，证明了椭圆曲线上量子化的唯一性，并证明了椭圆曲线上的Virasoro约束.第二篇论文（arXiv：1112.4063）证明了量子BCOV理论在椭圆曲线上的分拆函数与镜像椭圆曲线上的Gromov-Witten理论是等价的。第三篇论文，正在进行中，构造量子BCOV理论上的一般卡-丘。
Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we give examples of equations of surfaces of type $c_0+c_1x+c_2x^2+c_3x^3+c_4x^4=(ap^2+bq^2)^2$, $a,b,c \in \Q$ where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type $Y^2=f(a p^2+b q^2)$ where the polynomial $f$ is of degree $3$.
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