Do Sums of Squares Dream of Free Resolutions?

TL;DR

This paper explores the relationship between convex cones associated with real projective varieties and their algebraic properties, revealing how convex geometric features reflect homological conditions and generalizing previous results on sums of squares.

Contribution

It establishes a deep connection between the convex geometry of spectrahedral cones and the homological properties of projective varieties, extending prior work to reduced schemes and classifying spectrahedral cones with rank-one extreme rays.

Findings

Extreme rays of the dual cone have rank one iff the variety has Castelnuovo-Mumford regularity two.

Presence of higher rank extreme rays indicates failure of property N_{2,p}.

Results apply to matrix completion and moment problems on projective varieties.

Abstract

We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $\Sigma_X$ of sums of squares of linear forms. The dual cone $\Sigma_X^\ast$ is a spectrahedron and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $\Sigma_X^\ast$ have rank one if and only if X has Castelnuovo-Mumford regularity two. More generally, if $\Sigma_X^\ast$ has an extreme ray of rank $p > 1$, then $X$ does not satisfy the property $N_{2,p}$. We show that the converse also holds in a wide variety of situations: the smallest $p$ for which property $N_{2,p}$ does not hold is equal to the smallest rank of an extreme ray of $\Sigma_X^\ast$ greater than one. These results allow us to generalize the work of Blekherman-Smith-Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and to classify all spectrahedral cones with only rank one extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.