On Hermitian varieties in $\mathrm{PG}(6,q^2)$

$\mathbb{P}^{4}(\mathbb{F}_q)$中二次和非退化厄米簇的2次形式定义的码

Codes defined by forms of degree 2 on quadric and non-degenerate hermitian varieties in $\mathbb{P}^{4}(\mathbb{F}_q)$}

本文研究了G. Lachaud关于$\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$是秩为（$\mathcal{X}$）= 3，4，5的二次曲面或非退化埃尔米特簇.我们给出了$\mathcal{X}$的二次截面的点数%$# \mathcal{X}_{Z(\mathcal{Q})}(\mathbb{F}_{q})$的一些界，这些界是最好的，并且表明定义在非退化二次曲面上的码优于定义在退化二次曲面上的码。我们还展示了最小重量码字的几何结构，并估计了这些码的第二重量。对于非退化厄米簇$\mathcal{X}$，我们列出了前五个权重和相应的码字。文章最后以两个结论结束。一个是关于非奇异厄米簇$\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$上$h$阶泛函码的最小距离。关于非奇异厄米簇$\mathcal{X} \subset {\mathbb{P}}^N(\mathbb{F}_q)$上二阶泛函码前五个权码字分布的第二个猜想。

由 Hermitian 曲线得到的 PG(2,𝔽_{q^6}) 中的完全 (q+1)-弧

Complete $(q+1)$-arcs in $\mathrm{PG}(2,\mathbb{F}_{q^6})$ from the Hermitian curve

本文利用与弧相关的代数簇，证明当 q 足够大时 Hermitian 曲线的有理点在 PG(2,𝔽_{q^6}) 中构成完全 (q+1)-弧，解决了既有研究的开放情形。

低维中秩 $n$ Hermitian 簇和超曲面产生的函数码

Functional codes arising from rank $n$ Hermitian varieties and hypersurfaces in low dimensions

研究秩 $n$ 退化 Hermitian 簇 $P\\\mathcal{U}_{n-1}$ 在 $\\mathbb{P}^n(\\mathbb{F}_{q^2})$ 上的函数码 $C_d(\\mathcal{X})$，当 $d\\\leq q$ 时，通过建立交点数的上界，确定了 $n=2,3,4$ 时码的参数并刻画了达到最小距离的超曲面。

Deligne-Lusztig 变量、托里克轨道与 q-Klyachko 代数

Deligne--Lusztig varieties, toric orbifolds, and the $q$-Klyachko algebra

研究 q-Klyachko 代数的几何结构，当 q 为素数幂时，证明其为 Chow 环上的拉回映射像；当 q 为正有理数时，通过 toric 几何建立其 Kahler 包装。

$\mathrm{P G}\left(s, q^2\right)$ 中 $H\left(s,q^2\right)$ 非奇异超平面的刻画

Characterization of non-singular hyperplanes of $H\left(s,q^2\right)$ in $\mathrm{P G}\left(s, q^2\right)$

本文研究射影空间中非奇异埃尔米特簇的超平面刻画问题，通过分析超平面与点及余维2子空间的交点数，建立了超平面属于埃尔米特簇的充要条件，提供了一种纯组合识别方法。

On the $E$-polynomial of parabolic $\mathrm{Sp}_{2n}$-character varieties

We find the $E$-polynomials of a family of parabolic $\mathrm{Sp}_{2n}$-character varieties $\mathcal{M}^ξ_{n}$ of Riemann surfaces by constructing a stratification, proving that each stratum has polynomial count, applying a result of Katz regarding the counting functions, and finally adding up the resulting $E$-polynomials of the strata. To count the number of $\mathbb{F}_{q}$-points of the strata, we invoke a formula due to Frobenius. Our calculation make use of a formula for the evaluation of characters on semisimple elements coming from Deligne-Lusztig theory, applied to the character theo…

On Schubert varieties of complexity one

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.

Littlewood-Richardson coefficients via mirror symmetry for cluster varieties

I prove that the full Fock-Goncharov conjecture holds for $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})$-- the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential $W$ on $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})^\vee$ associated to the partial minimal model $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell}) \subset \mathrm{Conf}_3(\widetilde{\mathcal{F}\hspace{-1.6pt}…