[Paper Review] Grassmann defectivity à la Terracini

Carla Dionisi, Claudio Fontanari|ArXiv.org|Dec 14, 2001

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper establishes a correspondence between Grassmann defectivity of a variety $V$ and standard defectivity of its Segre product with $\mathbb{P}^k$, enabling the classification of Grassmann-defective varieties via classical Terracini-type methods. The key result is that the Veronese surface $V_{2,d}$ is not $(1,h)$-defective for $(d,h) \neq (3,4)$, with the exception of $V_{2,3}$ being $(1,4)$-defective, resolving a Waring-type problem for pairs of ternary forms.

ABSTRACT

This work is a modern revisitation of a classical paper by Alessandro Terracini, going back to 1915, which suggests an elementary but powerful method for studing Grassmann defective varieties. In particular, the case of Veronese surfaces is completely understood, giving an positive answer to the so-called Waring problem for pairs of homogeneous polynomials in three variables.

Motivation & Objective

• To generalize Terracini’s classical approach to Grassmann defectivity, extending his 1915 insights to a broader class of varieties.
• To establish a precise correspondence between $(k,h)$-Grassmann defectivity of a variety $V$ and $h$-defectivity of its Segre embedding $\sigma(\mathbb{P}^k \times V)$.
• To resolve the Waring-type problem for pairs of homogeneous polynomials in three variables by analyzing the defectivity of Veronese surfaces.
• To provide a complete classification of Grassmann defectivity for Veronese surfaces $V_{2,d}$, identifying the sole exception at $(d,h) = (3,4)$.

Proposed method

• Introduce the $(k,h)$-Grassmann secant variety $\operatorname{Sec}_{k,h}(V)$ as the closure of spans of $h+1$ independent points on $V$, each lying in a $k$-dimensional linear space.
• Prove that $V$ is $(k,h)$-defective with defect $\delta_{k,h}(V) = \delta$ if and only if $\sigma(\mathbb{P}^k \times V)$ is $h$-defective with $\delta_h(\sigma(\mathbb{P}^k \times V)) = \delta$, using the Segre embedding $\sigma$.
• Use the Jacobian criterion and local parametrization to analyze the dimension of the image of a multilinear map $\phi$ encoding the span of $h+1$ points and $k+1$ coefficients.
• Apply the Alexander-Hirschowitz theorem to show that linear systems $|\mathcal{L}_{2,d}(2^{h+1})|$ are nonspecial for $d \geq 5$, enabling dimension bounds.
• Analyze two cases: when the pencil of curves has a fixed component and when it does not, using Bertini’s theorem and base locus constraints.
• Derive inequalities from dimension constraints on linear systems and Grassmannians to derive contradictions unless $d=3,h=4$ is the only exception.

Experimental results

Research questions

• RQ1 Under what conditions is a Veronese surface $V_{2,d}$ $(1,h)$-Grassmann defective?
• RQ2 Can the Waring problem for pairs of ternary homogeneous polynomials of degree $d$ be solved using the same $h+1$ linear forms for all $k+1$ polynomials?
• RQ3 Is there a general correspondence between Grassmann defectivity of $V$ and standard defectivity of $\sigma(\mathbb{P}^k \times V)$?
• RQ4 What is the complete classification of $(k,h)$-defective Veronese surfaces $V_{2,d}$?
• RQ5 Why is the case $(d,h) = (3,4)$ the only exception to non-defectivity in the $(1,h)$-Grassmann setting?

Key findings

• The Veronese surface $V_{2,d}$ is not $(1,h)$-defective for any $d \neq 3$ or $h \neq 4$, meaning the expected dimension of the Grassmann secant variety is achieved.
• The only exception is $V_{2,3}$, which is $(1,4)$-defective, as confirmed by the existence of a pencil of plane cubics with a fixed conic component through five general points.
• The defectivity of $V_{2,3}$ as a $(1,4)$-Grassmann defective variety is equivalent to the $4$-defectivity of $\sigma(\mathbb{P}^1 \times V_{2,3})$, which has dimension 18 instead of the expected 19.
• For $d \geq 5$, the linear system $|\mathcal{L}_{2,d}(2^{h+1})|$ is nonspecial, so its dimension matches the expected dimension, enabling contradiction arguments.
• For $d=2$ and $d=3$, special linear systems are analyzed case by case, showing $\dim \mathcal{L}_{2,d}(2^{h+1}) = 0 < \delta/2$ when $\delta > 0$, contradicting the existence of $\delta$ independent pencils.
• The proof shows that no such configuration exists for $d=3,h=4$ except in the known exceptional case, confirming the sharpness of the theorem.