On the local version of the Severi problem

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 …

Some results of regularity for Severi varieties of projective surfaces

For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, δ}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with only $δ$ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper-bounds on $δ$ ensuring that the corresponding Severi variety is everywhere smooth of codimension $δ$ in $|C|$ (regular, for short). In particular, we focus on surfaces of general type, since for such surfaces less is known than what is proven for other case…

K-theory, LQEL manifolds and Severi varieties

We use topological K-theory to study non-singular varieties with quadratic entry locus. We thus obtain a new proof of Russo's Divisibility Property for locally quadratic entry locus manifolds. In particular we obtain a K-theoretic proof of Zak's theorem that the dimension of a Severi variety must be 2, 4, 8 or 16 and so resolve a conjecture of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.

A discretized Severi-type theorem with applications to harmonic analysis

In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of…

合成代数几何中的Chatelet定理

Chatelet's Theorem in Synthetic Algebraic Geometry

本文在合成代数几何框架下，针对任意基环证明了关于Severi-Brauer簇有理点的Chatelet定理的一个版本。

对数等纲平滑与最大节点退化

Logarithmic Equigeneric Smoothing and Maximal Nodal Degenerations

本文研究了曲线和曲面在代数族中的等纲变形问题，提出通过对数半正则性方法在退化情况下保持最大奇点数量的理论，并探讨了其在细化塞弗里计数中的应用。

曲线和三维流形中曲面的等奇异变形

Equisingular Deformations of Curves and Surfaces in Threefolds

本文研究曲线和曲面在三维流形中的等奇异变形问题，提出基于变形理论的标准，证明奇点数量由全局可实现性决定，并探讨其在退化情况下的应用。

椭圆曲线及其主齐性空间：分裂Severi--Brauer簇

Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties

本文研究哪些椭圆曲线可作为分裂Severi--Brauer簇的亏格一曲线的雅可比簇，给出三个新例子：刻画了完美域扩张下主齐性空间分裂Severi--Brauer簇的循环代数条件，构造了次数最小且不可再降的嵌入例子，并将Saltman关于曲面的结果推广到任意偶数维。

A local version of the Kawamata-Viehweg vanishing theorem

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

The local-global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois repres…

A local version of the Fulton-Hansen theorem relating secant and tangential varieties

Withdrawn by the authors, the main theorem is incorrect

Categorical Donaldson-Thomas theory for local surfaces: $\mathbb{Z}/2$-periodic version

We prove two kinds of $\mathbb{Z}/2$-periodic Koszul duality equivalences for triangulated categories of matrix factorizations associated with $(-1)$-shifted cotangents over quasi-smooth affine derived schemes. We use this result to define $\mathbb{Z}/2$-periodic version of Donaldson-Thomas categories for local surfaces, whose $\mathbb{C}^{\ast}$-equivariant version was introduced and developed in the author's previous paper. We compare $\mathbb{Z}/2$-periodic DT category with the $\mathbb{C}^{\ast}$-equivariant one, and deduce wall-crossing equivalences of $\mathbb{Z}/2$-periodic DT categorie…

算术Hodge指标定理在拟射影簇上的局部版本

A local version of the arithmetic Hodge index theorem over quasiprojective varieties

本文提出拟射影簇上度量线丛的局部相交数，证明局部算术Hodge指标定理，并由此得到Monge-Ampère方程在阿基米德与非阿基米德情形下的唯一性。