Do Sums of Squares Dream of Free Resolutions?

For a real projective variety $X$, the cone $\Sigma_X$ of sums of squares of linear forms plays a fundamental role in real algebraic geometry. The dual cone $\Sigma_X^*$ 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 $\Sigma_X^*$ have rank 1 if and only if $X$ has Castelnuovo--Mumford regularity two. More generally, if $\Sigma_X^*$ 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 $\Sigma_X^*$ greater than one. We generalize the work of Blekherman, Smith, and Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and classify all spectrahedral cones with only rank 1 extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.