Gap vectors of real projective varieties

TL;DR

This paper introduces the concept of gap vectors for real projective varieties, relating their properties to quadratic deficiencies, and provides formulas, bounds, and characterizations for these vectors, including explicit computations for Veronese embeddings.

Contribution

It systematically studies the properties of gap vectors, establishes a formula linking them to quadratic deficiencies, and characterizes varieties with minimal or simple gap vectors.

Findings

Gap vectors are weakly increasing.

Upper bounds for gap vector growth are proven and achieved.

Varieties of minimal degree have zero gap vectors.

Abstract

Let $X\subseteq \mathbb{P}^m$ be a totally real, non-degenerate, projective variety and let $\Gamma\subseteq X(\mathbb{R})$ be a generic set of points. Let $P$ be the cone of nonnegative quadratic forms on $X$ and let $\Sigma$ be the cone of sums of squares of linear forms. We examine the dimensions of the faces $P(\Gamma)$ and $\Sigma(\Gamma)$ consisting of forms in $P$ and $\Sigma$, which vanish on $\Gamma$. As the cardinality of the set $\Gamma$ varies in $1,\dots,\rm{codim}(X)$, the difference between the dimensions of $P(\Gamma)$ and $\Sigma(\Gamma)$ defines a numerical invariant of $X$, which we call the gap vector of X. In this article we begin a systematic study of its fundamental properties. Our main result is a formula relating the components of the gap vector of $X$ and the quadratic deficiencies of $X$ and its generic projections. The quadratic deficiency is a fundamental numerical invariant of projective varieties introduced by F. L. Zak. The relationship between quadratic deficiency and gap vectors allows us to effectively compute the gap vectors of concrete varieties as well as to prove several general properties. We prove that gap vectors are weakly increasing, obtain upper bounds for their rate of growth and prove that these upper bounds are eventually achieved for all varieties. Moreover, we give a characterization of the varieties with the simplest gap vectors: We prove that the gap vector vanishes identically precisely for varieties of minimal degree, and characterize the varieties whose gap vector equals $(0,\dots, 0,1)$. In particular, our results give a new proof of the theorem of Blekherman, Smith and Velasco saying that there are nonnegative quadratic forms which are not sums of squares on every variety, which is not of minimal degree. Finally, we determine the gap vector of all Veronese embeddings of $\mathbb{P}^2$, generalizing work of the first three authors.