Quadratic forms and singularities of genus one or two

Hyperelliptic involutions on generic normal surface singularities

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\tX$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\tX$ and investigate the existence of a complete linear se…

孤立超曲面奇点的谱亏格及一个与 Milnor 数相关的猜想

The spectral genus of an isolated hypersurface singularity and a conjecture relating to the Milnor number

本文提出孤立超曲面奇点的谱亏格概念，作为几何亏格的次要不变量，并预测其与 Milnor 数之间的不等式关系，在多种情形下验证了该猜想，且揭示了其与全纯解析挠渐近行为的深层联系。

具有给定奇点的K-微分

K-differentials with prescribed singularities

该研究探讨了黎曼曲面上k≥3的亚纯k-微分的局部不变量，证明了给定零点阶数模式下本原全纯k-微分的存在性，并揭示了不同亏格下k-留数配置的出现规律。

CR singularities of real fourfolds in $\mathbb{C}^3$

CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities of compact submanifolds.

Twin vector fields and independence of spectra for quadratic vector fields

The object of this paper is to address the following question: When is a polynomial vector field on $\mathbb{C}^2$ completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Moreover, we will see that we can always assume that both vector fields have the same singular locus and at each singularity both vector fields have the same spectrum. Let us say th…

亚纯 $\mathfrak{sl}_2(\mathbb{C})$-联络奇点的形变与亚纯二次微分

Deformations of singularities of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections and meromorphic quadratic differentials

本文研究无迹 $2\times2$ 亚纯线性 ODE 奇点的形变与汇合，建立形式不变量与二次微分射流的对应，并刻画等单值形变及其退化。