Characteristic numbers of rational curves with cusp or prescribed triple contact

Plane algebraic curves with prescribed singularities

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularit…

非结点实平面有理曲线的计数

Enumeration of non-nodal real plane rational curves

该研究探讨计数带非结点奇点的实有理平面曲线的不变量，证明唯一的此类不变量对应实有理三尖点四次曲线，且任意四对复共轭点的一般构型均存在一对这类曲线。

Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves

We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some …

Torsion properties of modified diagonal classes on triple products of modular curves

Consider three normalised cuspidal eigenforms of weight $2$ and prime level $p$. Under the assumption that the global root number of the associated triple product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel-Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associat…

Nodal curves with a contact-conic and Zariski pairs

In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

扭曲三重积根数与Darmon-Rotger循环

Twisted triple product root numbers and a cycle of Darmon-Rotger

研究三重积模曲线的代数循环，证明其为零循环，并探讨其在不同群作用下的性质，证明扭曲三重积L函数的全局根数为-1。

Enumerative geometry of plane curves of low genus

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety…

具有简单奇点的不可约平面曲线的亏量

Defect of irreducible plane curves with simple singularities

本研究围绕Dimca提出的平面曲线亏量展开，推导了含结点、普通尖点与普通三重点的不可约平面曲线的亏量下界，证明高Arnold指数的约化简单奇异平面曲线不可能是自由曲线。

计数 $\mathbb{P}^3$ 中具有退化奇点的平面曲线

Counting planar curves in $\mathbb{P}^3$ with degenerate singularities

本文针对 $\mathbb{P}^3$ 中位于 $\mathbb{P}^2$ 内、含 $δ$ 个结点和1个余维数 $k$ 奇点的 $d$ 次曲线数量问题，在 $δ+k \leq 4$ 时给出显式公式，通过扩展拓扑方法验证了与已有结果及相关猜想的一致性。

关于三重Hurwitz数结构的研究

Towards studying the structure of triple Hurwitz numbers

本文研究涉及三个非简单分支点的三重Hurwitz数，通过证明一个蕴含Frobenius恒等式的递推关系，获得了相关Hurwitz数的递推公式与显式公式，并详细探讨了单部拟三重Hurwitz数的降维公式、多项式性及与相交理论的联系。

Counting singular curves with tangencies

We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. Combined with the results of S. Basu and R. Mukherjee, this gives us a complete formula for the characteristic number of curves with $δ$-nodes and one singularity of type $A_k$, tangent to a given line, provided $δ+ k \leq 8$. We use a topological method, namely the method of dynamic intersections to compute the degenerate…

Sesquicuspidal曲线、scattering diagrams与辛非挤压

Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

本文解决了四维椭球体到四维球体的稳定化辛嵌入问题，通过建立定量辛几何与奇异代数曲线理论的桥梁并利用scattering diagrams框架，构造了新的有理代数曲线族并揭示了从Fibonacci阶梯到有理函数的相变。

Counting rational curves of arbitrary shape in projective spaces

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space.

On Hurwitz--Severi numbers

For a point $p\in CP^2$ and a triple $(g,d,\ell)$ of non-negative integers we define a {\em Hurwitz--Severi number} ${\mathfrak H}_{g,d,\ell}$ as the number of generic irreducible plane curves of genus $g$ and degree $d+\ell$ having an $\ell$-fold node at $p$ and at most ordinary nodes as singularities at the other points, such that the projection of the curve from $p$ has a prescribed set of local and remote tangents and lines passing through nodes. In the cases $d+\ell\ge g+2$ and $d+2\ell \ge g+2 > d+\ell$ we express the Hurwitz--Severi numbers via appropriate ordinary Hurwitz numbers. The …