Abstract. For a real projective variety X, the cone ΣX of sums of squares of linear forms plays a fundamental role in real algebraic geometry. The dual cone Σ * 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 Σ * X have rank 1 if and only if X has Castelnuovo-Mumford regularity two. More generally, if Σ * X has an extreme ray of rank p > 1, then X does not satisfy the property N2,p. We show that the converse also holds in a wide variety of situations: the smallest p for which property N2,p does not hold is equal to the smallest rank of an extreme ray of Σ * 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.Key words. sums of squares, spectrahedra, free resolutions, Castelnuovo-Mumford regularity AMS subject classifications. 14P05, 13D02, 52A99, 05C50 DOI. 10.1137/16M10845601. Introduction. Minimal free resolutions and spectrahedra are central objects of study in commutative algebra and convex geometry, respectively. We connect these disparate areas via real algebraic geometry and show a surprisingly strong connection between convexity properties of certain spectrahedra and the minimal free resolution of the defining ideal of the associated real variety. In the process, we address fundamental questions on the relationship between nonnegative polynomials and sums of squares.In real algebraic geometry, we associate two convex cones to a real projective variety X: the cone P X of quadratic forms that are nonnegative on X, and the cone Σ X of sums of squares of linear forms. A recent line of work shows that convexity properties of these cones are strongly related to geometric properties of the variety X over the complex numbers [2], [7], and [4]. We extend these novel connections into the realm of homological algebra by showing a direct link between the convex geometry of the dual convex cone Σ * X and property N 2,p of the defining ideal of X: for an integer p 1, the scheme X satisfies property N 2,p if the jth syzygy module of the homogeneous ideal of X is generated in degree j + 2 for all j < p.

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.

Spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices. We characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic.

Abstract. We study the real rank of points with respect to a real variety X. This is a generalization of various tensor ranks, where X is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of X only. We show constructively that there exist varieties X for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman-Teitler on the maximal X-rank is tight. We also give examples of varieties X for which the gap between maximal complex and the maximal real rank is arbitrarily large. To facilitate our constructions we prove a conjecture of Reznick on the maximal real symmetric rank of symmetric bivariate tensors. Finally we study the geometry of the set of points of maximal real rank in the case of real plane curves.

Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. This is a fundamental object in polynomial optimization and convex algebraic geometry. We summarize results on sums of squares that fit naturally into the context of Gram spectrahedra, present some new results, and highlight related open questions. We discuss sum-of-squares representations of minimal length and relate them to Hermitian Gram spectrahedra and point evaluations on toric varieties.