Point Configurations and Cayley-Menger Varieties

射影托利丘变种中的有理曲线

Rational Curves in Projective Toric Varieties

本文研究了射影托利丘变种中有理曲线的构造与性质，提出度数为d的凯莱结构，并建立了希尔伯特方案与最大光滑凯莱结构之间的对应关系。

Cayley-Hamilton元组：代数簇与联合谱之间的相互作用

Cayley--Hamilton tuples: an interplay between algebraic varieties and joint spectra

本文引入了Cayley-Hamilton元组的概念，即满足某种多项式 annihilating 的交换算子元组，并探讨了其与代数簇的关系。通过构造无限维情形下的两个元组家族，揭示了多项式零点与联合谱的联系，为相关数学领域提供了新的视角。

Cayley forms and self dual varieties

Generalized Chow forms were introduced by Cayley for the case of 3-space, their zero set on the Grassmannian G(1,3) is either the set Z of lines touching a given space curve (the case of a `honest' Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1,3). Our main result is that F is a Cayley form if and only if Z = G(1,3) \cap {F=0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quad…

The double Cayley Grassmannian

We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type G2), but doubled in the certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.

On intersections of symmetric determinantal varieties and theta characteristics of canonical curves

From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…

亏格四曲线的 Prym 簇

Prym varieties of genus four curves

本文建立亏格四曲线二重覆盖与 Cayley 三次曲面的对应，将相关 Prym 簇描述为亏格三曲线 Jacobian 的二次扭曲，并证明该构造在模空间特定除子补集上的延拓。

Codimension one foliations on homogeneous varieties

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such resu…

Geometric Equations for Matroid Varieties

Each point $x$ in Gr$(r,n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in \mathrm{Gr}(r,n) | M_y = M_x\}$ form a stratification of Gr$(r,n)$ with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals $I_x$ of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective g…

Generalizing the Converse to Pascal's Theorem via Hyperplane Arrangements and the Cayley-Bacharach Theorem

Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points and if we color the remaining lines so that each triple point lies on a red and blue line then the points of intersection of the red and blue lines lying off the green line lie on a unique curve of degree k-1. We also use these ideas to extend a second generalization of the Braikenridge-Maclaurin Theorem, due to Möbius. Finally we use Terrac…

对称星变换及其对偶微分算子的代数几何

Symmetric star transforms and the algebraic geometry of their dual differential operators

本文研究星变换的对偶微分算子代数几何性质，证明对称分支向量下算子象征属于不变多项式环，退化对称星变换对应初等对称多项式零点集的线性子空间，二维不可逆星变换对应Cayley结点三次曲面的实直线。

仙人掌、Pascal 与 Pappus 拟阵的代数几何

Algebraic Geometry of Cactus, Pascal, and Pappus Matroids

本文研究秩三点—线构型拟阵簇的定义理想，利用可提升性与 Grassmann–Cayley 代数给出 Pascal、Pappus 及仙人掌拟阵理想的完整有限生成集，并证明仙人掌拟阵可实现且其簇不可约。

On the infinitesimal rigidity of homogeneous varieties

Let $X\subset P^N$ be a variety (respectively a patch of an analytic submanifold) and let $x\in X$ be a general point. We show that if the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Segre $P^n\times P^m$, $n,m\geq 2$, a Grassmaniann $G(2,n+2)$, $n\geq 4$, or the Cayley plane $OP^2$, then $X$ is the corresponding homogeneous variety (resp. a patch of the corresponding homogeneous variety). If the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Veronese $v_2(P^n)$ and …