Linear systems in $\mathbb{P}^2$ with base points of bounded multiplicity

Linear systems over P1xP1 with base points of multiplicity bounded by three

We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities do not exceed 3.

Intersection multiplicity of a sparse curve and a low-degree curve

Let $F(x, y) \in \mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) \in \mathbb{C}[x,y]$ be a polynomial with $t$ monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x,y) = G(x,y) = 0$. Our main result is that the multiplicity of any isolated solution $(a,b) \in \mathbb{C}^2$ with nonzero coordinates is no greater than $\frac{5}{2}d^2t^2$. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of $F$ and $G$, and we briefly review some connections between sparse polynomials and algebraic complexity the…

Local Properties of Richardson Varieties in the Grassmannian via a Bounded Robinson-Schensted-Knuth Correspondence

We give an explicit Grobner basis for the ideal of the tangent cone at any T-fixed point of a Richardson variety in the Grassmannian, thus generalizing a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Grobner basis result to deduce a formula which computes the multiplicity of the Richardson variety at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattehthaler.

Special homogeneous linear systems on Hirzebruch surfaces

The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces F_n. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.

由环簇决定的实有界重数非结式组的空间

Spaces of non-resultant systems of real bounded multiplicity determined by a toric variety

本文定义与环簇扇相关的实有界重数非结式组空间，证明其同伦稳定性并显式给出稳定维数。

柯代拉-伊塔卡维度与乘数：一种分析视角

Kodaira-Iitaka dimension and multiplicity: an analytic perspective

本文从分析角度研究了graded线性序列的柯代拉-伊塔卡维度与乘数，通过 plurisubharmonic 包络的交理论得到改进的公式，并引入非 pluripolar 的数值柯代拉-伊塔卡维度。