Abstract. Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

Abstract. We study the geometry underlying the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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Abstract. The main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the Fano variety of lines of a cubic 4-fold F , in general different from F = Z(f ).A general K3 surface S of genus 8 determines uniquely a pair of cubic 4-folds: the apolar cubic F (S) and the dual Pfaffian cubic F (S) (or for simplicity F and F ). As Beauville and Donagi have shown, the Fano variety F F of lines on the cubic F is isomorphic to the Hilbert scheme Hilb 2 S of length two subschemes of S. The first main result of this paper is that Hilb 2 S parametrizes the variety V SP (F, 10) of presentations of the cubic form f , with F = Z(f ), as a sum of 10 cubes, which yields an isomorphism between F F and V SP (F, 10). Furthermore, we show that V SP (F, 10) sets up a (6, 10) correspondence between F and F F . The main result follows by a deformation argument.1. Pfaffian and apolar cubic 4-folds associated to K3 surfaces of genus 8 1.1. Let V be a 6-dimensional vector space over C. Fix a basis e 0 , . . . , e 5 for V ; then e i ∧ e j for 0 ≤ i < j ≤ 5 form a basis for the Plücker space ∧ 2 V of 2-dimensional subspaces in V or lines in P 5 = P(V ). We associate to a 2-vector g = i

We give a lower bound for the degree of a finite apolar subscheme of a symmetric form F , in terms of the degrees of the generators of the annihilator ideal F ⊥ . In the special case, when F is a monomial. . ≤ d n−1 ≤ dn we deduce that the minimal length of an apolar subscheme of F is (d 0 + 1) · . . . · (d n−1 + 1), and if d 0 = . . . = dn, then this minimal length coincides with the rank of F .