Nontrivial linear projections of the Grassmannian Gr_3(C^6)

Injectivity of generalized Wronski maps

We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

三次Fano三维簇与14次Fano三维簇的稳定等变双有理等价

Stable equivariant birationalities of cubic and degree 14 Fano threefolds

本研究提出Pfaffian-Grassmannian对应的等变版本，以此构造出三次三维簇与14次Fano三维簇间非平凡的扭稳定等变双有理等价实例，为等变双有理几何提供新案例。

Degrees of projections of rank loci

We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stem from applications to `matrix rigidity'; we also view them as an excellent source of examples to test methods in intersection theory, particularly computations of Segre classes. Our results are generally expressed in terms of intersection numbers in Grassmannians, which can be computed explicitly in many cases. We observe that, surprisingly (to us), these degrees appear to match the numbers of …

The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification of the real Grassmannian in order to characterize the unbounded line-solitons in the contour …

Chow-Lam 恢复

Chow-Lam Recovery

本文研究 Grassmannian 子簇能否由其 Chow-Lam 形式及线性投影恢复，给出恢复的必要条件，并构造与射影空间情形相反、无法恢复的反例族。

作为仿射簇的 Grassmann 簇的次数

Degree of the Grassmannian as an affine variety

本文研究应用数学中投影矩阵与对合矩阵嵌入下 Grassmann 簇的仿射次数，给出显式公式，并证明及推广相关猜想。

Affine Hermitian Grassmann Codes

The Grassmannian is an important object in Algebraic Geometry. One of the many techniques used to study the Grassmannian is to build a vector space from its points in the projective embedding and study the properties of the resulting linear code. We introduce a new class of linear codes, called Affine Hermitian Grassman Codes. These codes are the linear codes resulting from an affine part of the projection of the Polar Hermitian Grassmann codes. They combine Polar Hermitian Grassmann codes and Affine Grassmann codes. We will determine the parameters of these codes.

Galois subspaces for the rational normal curve

We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as a disjoint union of locally closed subvarieties in the Grassmannian $\mathbb{G}(n-2,n)$.

短旗簇的热带化与退化及其纤维

Tropicalization and degeneration of short flag varieties and their fibers

本文研究线性退化短旗簇的热带化，利用拟阵方法证明其纤维在温和假设下为线性空间，从而给出其热带化的完整组合描述。

Probabilistic Schubert Calculus

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\textrm{edeg} G(k,n)$ of the real Grassmannian $G(k,n)$ as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random subspaces of $\mathbb{R}^n$, all of dimension $n-k$, where these subspaces are sampled uniformly and independently from $G(n-k,n)$. We express $\textrm{edeg} G(k,n)$ in terms of the volume of an invariant convex body in the tangent space to the Grassmanian, and prove that for fixed $k\ge 2$ and $n\to\infty$, $$ \textrm{edeg} G(k,n) = \textrm{deg} G_\mathbb{…

具有特殊投影性质的 $\mathbb{P}^4$ 中有限点集

Finite sets of points in $\mathbb{P}^4$ with special projection properties

本文提出并研究 $\mathbb{P}^4$ 中 $(b,d)$-geprofi 点集，证明其非平凡存在当且仅当 $b\geq4$、$d\geq2$，并发现无穷多参数下存在线性一般位置的此类集合。