Hyperplane sections of Legendrian subvarieties

On Legendrian curves in $\mathbb P^3$

We show that if a smooth projective curve $C\subset\mathbb P^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$ is unique up to a linear automorphism) and $C$ is linearly normal (i.e., not an isomorphic linear projection of a smooth curve $C'\subset\mathbb P^n$, $n>3$, where $C'$ does not lie in a hyperplane) then $C$ is a twisted cubic or a line.

关于射影空间中某些子簇余集上整点的潜在密度的研究

On potential density of integral points on the complement of some subvarieties in the projective space

本文研究了数域上 $n$ 维射影空间中特定闭子簇余集上整点的潜在密度，推广了 Corvaja-Zucconi 在低维射影空间中的相关结果。

Canonical map of low codimensional subvarieties

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq λd^{ε_1}+c(\sum_{1\leq h < ε_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ o…

Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights.…

幂零轨道闭包上的Legendrian直线族

Legendrian families of lines on nilpotent orbit closures

该研究针对复半单李代数的幂零轨道闭包，分析过一般点的直线切方向空间的不可约分量性质，在两类情形下研究其Legendrian分量，并给出相关推论。

来自Gushel-Mukai四维簇的Bridgeland模空间的Lagrangian族

Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

本文证明Gushel-Mukai簇与其超平面截面的Kuznetsov分量间拉回与推出函子保持Bridgeland稳定性，并由此构造GM四维簇模空间的8维Lagrangian族及证明GM三维簇模空间的射影性。

具有简单正规交叉点的代数簇的投射光滑化

Projective smoothing of varieties with simple normal crossings

该研究提出基于态射形变理论与ribbons概形的新方法，证明投射空间中带两个Fano分支的简单正规交叉簇的存在性与光滑化，给出三维簇的显式构造及对应slc对的光滑化结果。