[Paper Review] On the Alexander-Hirschowitz Theorem

Maria Chiara Brambilla, Giorgio Ottaviani|ArXiv.org|Jan 15, 2007

Tensor decomposition and applications17 references4 citations

TL;DR

This paper presents a self-contained, simplified proof of the Alexander-Hirschowitz theorem, which determines when a general collection of $k$ double points in $\mathbb{P}^n$ imposes independent conditions on degree-$d$ homogeneous polynomials, with a well-known list of exceptions. The proof builds on Terracini’s lemmas and the differential Horace method, with new simplifications—especially for the cubic case ($d=3$)—and provides a comprehensive historical overview of the problem’s resolution.

ABSTRACT

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case $d=3$, where our proof is shorter. We end with an account of the history of the work on this problem.

Motivation & Objective

• To provide a self-contained and accessible proof of the Alexander-Hirschowitz theorem, which resolves a classical problem in algebraic geometry concerning the dimension of higher secant varieties.
• To simplify and clarify the original proof by Alexander and Hirschowitz, particularly by streamlining the argument for the cubic case ($d=3$), which is claimed to be shorter than prior treatments.
• To unify and present the historical development of the problem, from early conjectures by Palatini and Terracini to the final resolution by Alexander and Hirschowitz in 1995.
• To establish the equivalence between the condition on linear systems with double points and the expected dimension of higher secant varieties of Veronese embeddings.
• To highlight the role of degeneration techniques, zero-dimensional schemes, and the differential Horace method in resolving the problem, especially in the critical case $d=3$.

Proposed method

• Utilizes Terracini’s two fundamental lemmas on the tangent spaces of Veronese varieties and the behavior of linear systems with multiple points.
• Applies the differential Horace method, a technique developed by Alexander and Hirschowitz, to control the dimension of linear systems via inductive degeneration arguments.
• Employs degeneration to curvilinear schemes and the use of zero-dimensional schemes to analyze the failure of linear systems to impose independent conditions.
• Incorporates Chandler’s Curvilinear Lemma (Lemma 6.1) to simplify the analysis for $d \geq 4$, and provides a detailed treatment of the $d=3$ case with new simplifications.
• Uses semicontinuity and the structure of the Veronese embedding $V^{d,n} \subset \mathbb{P}^m$ to relate the expected codimension of $I_X(d)$ to the dimension of $\sigma_k(V^{d,n})$.
• Relies on the duality between homogeneous polynomials and symmetric tensors, and the natural pairing $S^dV \otimes S^dV^\vee \to \mathbb{K}$, to define vanishing conditions at double points.

Experimental results

Research questions

• RQ1 Under what conditions does a general collection of $k$ double points in $\mathbb{P}^n$ impose independent conditions on the space of degree-$d$ homogeneous polynomials?
• RQ2 Why do the exceptional cases—such as $n=2, d=4, k=5$ or $n=4, d=3, k=7$—fail to satisfy the expected codimension?
• RQ3 How can the differential Horace method be systematically applied to prove the expected dimension of higher secant varieties of Veronese embeddings?
• RQ4 What simplifications can be achieved in the proof of the Alexander-Hirschowitz theorem, especially in the case $d=3$, using modern techniques and insights?
• RQ5 To what extent can combinatorial or tropical-geometric methods, such as degenerating the Veronese surface into planes, be generalized to higher-dimensional Veronese varieties?

Key findings

• The Alexander-Hirschowitz theorem holds with the expected codimension $\min\left((n+1)k, \binom{n+d}{n}\right)$ for all $k$ and $d$, except for five specific exceptional cases.
• The exceptional cases are: $d=2$, $2 \leq k \leq n$; $n=2$, $d=4$, $k=5$; $n=3$, $d=4$, $k=9$; $n=4$, $d=3$, $k=7$; $n=4$, $d=4$, $k=14$.
• For $d=3$, the authors present a significantly shorter and more streamlined proof than previous treatments, marking a key original contribution of the paper.
• The equivalence between the linear system condition (vanishing of first derivatives at double points) and the expected dimension of the $k$-secant variety $\sigma_k(V^{d,n})$ is rigorously established over algebraically closed fields of characteristic zero.
• The proof is valid for $\operatorname{char}(\mathbb{K}) = 0$, and the equivalence between the two formulations (ideal-theoretic and secant variety) holds under this assumption.
• The work confirms that the Veronese variety $V^{d,n}$ is one of the few classes of varieties for which the dimension of all higher secant varieties is completely known, despite the complexity of the problem.