On the Alexander-Hirschowitz Theorem

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…

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…

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).

On the Hartshorne-Hirschowitz theorem

The Hartshorne--Hirschowitz theorem says that a generic union of lines in $\mathbb{P}^n$, $(n\geq 3)$, has good postulation. The proof of Hartshorne and Hirschowitz in the initial case $\mathbb{P}^3$ is difficult and so long, which is handled by a method of specialization via a smooth quadric surface with the property of having two rulings of skew lines. We provide a proof in the case $\mathbb{P}^3$ based on a new degeneration of disjoint lines via a plane $H\cong\mathbb{P}^2$, which we call $(2,s)$-cone configuration, that is a schematic union of $s$ intersecting lines passing through a singl…

An invariant regarding Waring's problem for cubic polynomials

We compute the equation of the 7-secant variety to the Veronese variety (P^4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariantof plane cubics as a pfaffian.

调和多项式的各向同性秩

Isotropic Rank of Harmonic Polynomials

本文研究了调和多项式的各向同性秩，通过分类secant种群的维度，解决了三次形式、二次曲面和单项式的各向同性秩问题，并类比Alexander-Hirschowitz定理。

Twisted Alexander polynomials of Plane Algebraic Curves

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degeneration…

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。

Rosenberg's Reconstruction Theorem (after Gabber)

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its abelian category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber.

Some analogs of Zariski's Theorem on nodal line arrangements

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexan…

扭结同调跳跃点集、扭结亚历山大多项式与$\Sigma$-不变量

Twisted homology jump loci, twisted Alexander polynomials, and $Σ$-invariants

本文引入扭结同调跳跃点集作为新的不变量族，通过热带几何得到BNSR $\Sigma$-不变量的更优上界，并证明在紧致可定向3-流形和紧致Kähler流形上该上界是紧的，且给出比经典未扭结情形更强的几何可实现性障碍。