Macdonald formula for curves with planar singularities

从平面曲线奇点的计数几何角度看对数凹性

Log-concavity from enumerative geometry of planar curve singularities

本文提出关于平面曲线奇点计数几何中BPS不变量的对数凹性猜想，并将其扩展到Legendrian链环的规则多项式和特征簇的E多项式，证明了不可约加权齐次奇点（扭线结）和ADE奇点的猜想，以及规则多项式的乘法性质。

平面慢度曲面的奇点分类

Classification of singularities of planar slowness surfaces

本文完全分类了二维弹性波慢度曲面的奇点类型，包括横截自交和同心圆与椭圆相切产生的切向奇点，并证明了横截自交时弹性波算子的主符号局部光滑可对角化。

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。

Invariants of plane curve singularities and Newton diagrams

We present an intersection-theoretical approach to the invariants of plane curve singularities $μ$, $δ$, $r$ related by the Milnor formula $2δ=μ+r-1$. Using Newton transformations we give formulae for $μ$, $δ$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities.

Autoduality of compactified Jacobians for curves with plane singularities

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

The canonical ring of a 3-connected curve

Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, ω_C)= \oplus_{k \geq 0} H^0(C, ω_C^k is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L)=\oplus_{k \geq 0} H^0(C,L^k)$ is generated in degree 1 if C is reduced with planar singularities and L is an invertible sheaf such that deg L_{|B} \geq 2p_a(B)+1 for every B \subseteq C.

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Mapl…

整合法平面曲线上点的希尔伯特概型的半正交不可分解性

Semiorthogonal indecomposability for Hilbert schemes of points on integral locally planar curves

该研究针对代数闭域上具局部平面奇点的整射影曲线的点希尔伯特概型，证明其相关范畴的半正交不可分解性，建立了相对结论，结果适用于任意特征。

Curve Singularities and Monster / Semple Towers

In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact pseudogroup on the tower. Here, we construct the analogous tower for curves in $n$-space. (This tower is known as the Semple Bundle in Algebraic Geometry.) The pseudo-group of diffeomorphisms of $n$-space acts on each level of the extended tower. We take initial steps toward classifying points of this extended Monster tower under this pseudogroup action. Arnol…

瞬子切片及其超多项式

Instanton slices and their superpolynomials

研究任意秩平面曲线奇点的动机超多项式与瞬子切片超多项式等的联系，通过平面曲线奇点适当族超多项式的归纳极限得瞬子和新公式，有望影响相关领域，还对特定双曲结超多项式有动机解释及相关猜想。

关于具有孤立奇点的某些曲线和曲面的总Tjurina数的组合界

On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

本文研究仅具有普通拟齐次奇点的平面线-圆锥曲线排列，导出多重交点分布的结构不等式，从而建立自由线排列和自由圆锥曲线排列的总Tjurina数的下界，并推广到三维空间中的超曲面。

Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical cas…

非预期Stein填充、有理曲面奇点与平面曲线配置

Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

针对约化基本闭链的有理曲面奇点，研究构建辛几何类比构造，通过辛圆盘配置描述其链环的所有Stein填充，证明部分奇点存在非Milnor纤维的Stein填充，同时给出极小消解自交数≤-5时Stein填充唯一的结论。