Geometry of reduced sextics of torus type

Fundamental groups of symmetric sextics

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type $\bold{E}_6$ singular points. As a simple application, we compute the fundamental groups of 125 other sextics, most of which are new.

A remark on a Nagell-Lutz type statement for the Jacobian of a curve of genus 2 and a (2, 3, 6) quasi-torus decomposition of a sextic with 9 cusps

In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson Conjecture

Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we study the configuration of points of certain reduced zero dimensional subschemes on $S$ satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small $c_2$. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components …

Singular cubic fourfolds containing a plane

In this paper we consider cubic 4-folds containing a plane whose discriminant curve is a reduced nodal plane sextic. In particular, we describe the singular points of such cubic 4-folds and we give an estimate of the rank of the free abelian group generated by the equivalence classes of the algebraic cycles of codimension 2. Moreover, we describe some conditions on the geometry of the plane sextics so that all the associated cubic 4-folds are singular and we construct a family of smooth rational cubic 4-folds whose discriminant curve is reduced but reducible.

Standard vs. Reduced Genus-One Gromov-Witten Invariants

We give an explicit formula for the difference between the standard and reduced genus-one Gromov-Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi-Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case.

DAHA and iterated torus knots

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the O…

约化Čech复形与环面映射下高次直像的计算

Reduced Čech complexes and computing higher direct images under toric maps

本文提出约化Čech复形的公理化表述与环面簇上的构造方法，并给出光滑真环面簇间环面纤维化下线丛高次直像的计算算法，解决层上同调计算与直像求解问题。

Jacobian varieties of reduced tropical curves

On tropical geometry in $\rr^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if the bunch is a bouquet, then the Jacobian is a higher-dimensional torus.

Vanishing theorems for toric polyhedra

A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic. Finally, we give a very powerful extension theorem for ample line bundles.

Reduced phase space and toric variety coordinatizations of Delzant spaces

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coord…

基于环面作用的有理曲线计数

Enumeration of rational curves via torus actions

本文包含了一个尝试，制定严格的，并检查预测枚举几何的曲线以下镜像对称。主要工具是一个新的稳定映射的概念。本文给出了所有代数射影或闭辛流形的Gromov-Witten不变量的一个轮廓。镜像对称在五次三重有理曲线的基本例子中被简化为某些复杂但明确的恒等式。计算策略如下：1）将计数问题归结为关于周围射影空间上曲线空间上的Chern类的问题; 2）利用Bott剩余公式，将其传递到在对角矩阵群作用下不变的（退化）曲线空间;我们的计算方案给出了“封闭”公式的生成函数的拓扑sigma模型的广泛的一类流形，涵盖了许多Calabi-Yau和Fano品种。

矩阵 Schubert 簇、二项式理想与约化 Gröbner 基

Matrix Schubert varieties, binomial ideals, and reduced Gröbner bases

本文研究 Schubert 行列式理想在 Knutson-Miller 项序下的约化 Gröbner 基项数下界，并据此刻画二项式矩阵 Schubert 簇、统一正则度公式及判定 Gao-Yong 基何时约化。

On the central geometry of nonnoetherian dimer algebras

Let $Z$ be the center of a nonnoetherian dimer algebra on a torus. Although $Z$ itself is also nonnoetherian, we show that it has Krull dimension $3$, and is locally noetherian on an open dense set of $\operatorname{Max}Z$. Furthermore, we show that the reduced center $Z/\operatorname{nil}Z$ is depicted by a Gorenstein singularity, and contains precisely one closed point of positive geometric dimension.