An asymptotic Alexander-Hirschowitz theorem for surfaces

Computing limit linear series with infinitesimal methods

Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points which approach a divisor. On the other hand, Nagata in connection with its counter example to the fourteenth problem of Hilbert determined the Hilbert function H(d) of the union of k^2 points of the same multiplicity m in the plane up to degree d=km. We introduce a new method to determine…

3-dimensional sundials

Robin Hartshorne and Alexander Hirschowitz proved that a generic collection of lines on $\mathbb P^n$, $n \geq 3$, has bipolynomial Hilbert Function. We extended this result to a specialization of the collection of generic lines, by considering a union of lines and $3$-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).

加权射影空间中的插值

Interpolation in Weighted Projective Spaces

本文利用交换代数方法研究加权射影空间中的双重插值问题，证明一般简单点的 Hilbert 函数具有预期行为，并通过归纳程序和 Terracini 引理给出 Alexander-Hirschowitz 定理在加权射影平面中的无例外情形及无穷族插值下界。

Secant varieties to osculating varieties of Veronese embeddings of $\mathbb{P}^n$

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\mathbb{P}^n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity…

Dimension of linear systems: a combinatorial and differential approach

We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension of linear systems of hypersurfaces in a projective space $\PP^n$ with generically prescribed singularities, and the calculus of collisions of fat points in $\PP^2$. These applications will be treated independently but a simple example in the introduction explains how the theorem will be used.

On the secant varieties of tangential varieties

Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Let $σ_{a,b}(X)\subseteq \mathbb{P}^r$, $(a,b)\in \mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $\dim σ_{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $\mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $\dim σ_{0,b}(X)$ is the expected o…

关于部分多项式插值

On partial polynomial interpolation

本文将Alexander-Hirschowitz定理推广到包含在一般双点并集中的任意零维方案，在多项式插值框架下证明了除d=2的五个例外情况外，给定任意多个一般一阶偏导线性组合值的次数≤d的多项式仿射空间具有预期维数。

具有普通双点的射影超曲面的伯恩斯坦 - 佐藤多项式的根

Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points

研究具有普通双点的射影超曲面的伯恩斯坦 - 佐藤多项式的根，给出根的相关形式及\(p_f\)的上界，通过构造齐次多项式证明特定情况下上界精确，并提及相关猜想。

复光滑拟射影簇的 L^2 型不变量——综述

$L^2$-type invariants for complex smooth quasi-projective varieties -- a survey

本文综述了复光滑拟射影簇在满同态到 Z^n 下有限阿贝尔覆盖的 L^2 型不变量研究，建立其与 Alexander 不变量及上同调跳跃轨迹的联系，并在 orbifold 有效情形给出一次同调的显式公式，同时提出超平面排列补集的相关公开问题。