[Paper Review] Osculating Varieties of Veronesean and their higher secant varieties

Alessandra Bernardi, Maria Virgina Catalisano+2 more|ArXiv.org|Mar 8, 2004

Tensor decomposition and applications11 references18 citations

TL;DR

This paper investigates the dimension of higher secant varieties to osculating varieties of Veronese embeddings using inverse systems and apolarity theory. It establishes conditions under which these secant varieties are defective, providing explicit lower bounds on defectivity based on combinatorial properties of forms and fat point schemes.

ABSTRACT

We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension of their higher secant varieties. Via inverse systems (apolarity) and the study of certain spaces of forms we are able, in several cases, to determine whether those secant varieties are defective or not.

Motivation & Objective

• To determine when higher secant varieties to osculating varieties of Veronese embeddings are defective.
• To characterize the defectivity of $O_{k,n,d}^s$, the $(s-1)$-st higher secant variety of the $k$-th osculating variety of the Veronese embedding $X_{n,d}$.
• To apply inverse systems and apolarity techniques to analyze the dimension of linear spans of osculating spaces.
• To establish quantitative lower bounds on the defect of $O_{k,n,d}^s$ in terms of binomial coefficients related to form degrees and variables.
• To conjecture that defectivity arises only when fat point schemes impose unexpected conditions, linking geometric defectivity to postulation problems.

Proposed method

• Uses Terracini’s Lemma to relate the dimension of $O_{k,n,d}^s$ to the linear span of tangent spaces at generic points.
• Applies the duality between forms and ideals via inverse systems (apolarity) to study linear dependence in the span of osculating spaces.
• Analyzes the vector space $W_1 + igcdots + W_s = ig\<x_0^\beta R_k, x_0^{\beta-1}F_1 R_1, \dots, x_{s-1}^\beta R_k, x_{s-1}^{\beta-1}F_s R_1\big\rangle$, where $\beta = d - k$, to compute expected vs. actual dimensions.
• Employs combinatorial counting of monomials in $R_d$ not contained in the span to detect defectivity, especially in cases where $s \leq n$ or $s = n+1$.
• Leverages the structure of $s$-tuples of $k$-fat points and their ideals $\mathcal{I}_{Z,X} \otimes \mathcal{L}$ to compute cohomological dimensions and defect via $h^0$-vanishing.
• Derives defect bounds by identifying independent relations among generators of the span, particularly from intersections like $x_i^\beta R_k \cap x_i^{\beta-1}F_{i+1}R_1$ and cross-terms $x_i^\beta x_j^\beta F$ for $i \neq j$.

Experimental results

Research questions

• RQ1 When is the $s$-th higher secant variety $O_{k,n,d}^s$ of the $k$-th osculating variety of the Veronese embedding $X_{n,d}$ defective?
• RQ2 What are the precise combinatorial conditions on $n$, $d$, $k$, and $s$ that lead to defectivity in $O_{k,n,d}^s$?
• RQ3 How do the postulation properties of fat point schemes relate to the defectivity of $O_{k,n,d}^s$?
• RQ4 Can the defect of $O_{k,n,d}^s$ be bounded below using binomial coefficients derived from the degrees and number of variables?
• RQ5 Under what conditions does $O_{k,n,d}^s$ fail to achieve its expected dimension, and what causes this failure?

Key findings

• For $s \leq n$ and $k+2 \leq d \leq 2k$, the variety $O_{k,n,d}^s$ is defective with defect $\delta \geq \binom{n-s+d}{d}$ when the expected dimension is $\binom{d+n}{n}$.
• When $s \leq n$ and the expected dimension is $s\binom{k+n}{n} + sn$, the defect is at least $\binom{s}{2} \binom{2k-d+n}{n}$, arising from independent relations among $x_i^\beta x_j^\beta F$ terms.
• For $s = n+1$, $k+2 \leq d \leq 2k$, and expected dimension $ (n+1)\left(\binom{k+n}{n} + n\right) $, the defect satisfies $\delta \geq \binom{n+1}{2} \binom{2k-d+n}{n}$.
• When $s = n+1$, $n \geq \frac{k+2}{d-k-2}$, and expected dimension is $N = \binom{d+n}{n}$, the defect is at least $\binom{(n+1)(d-k-1)-(d+1)}{n}$, derived from counting monomials of degree $d$ with bounded exponents.
• The defect arises from non-trivial relations among generators of the span, particularly from intersections of ideals associated with fat points and from symmetric monomials in $R_d$ not lying in the span.
• The authors conjecture that defectivity of $O_{k,n,d}^s$ occurs only when the postulation of fat point schemes $X$ or $T$ imposes unexpected conditions, linking geometric defectivity to algebraic postulation problems.