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.

We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.

We study maximal families W of the Hilbert scheme, H(d, g) sc , of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g) sc and we determine dim W . If s = 4 and W is an irreducible component of H(d, g) sc , then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s − 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g) sc , thus making progress in proving a conjecture for such families.

Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of the ten nodes of the corresponding real quartic surface. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli Theorem for real K3 surfaces. We here give a new proof that is self-contained and algorithmic. This involves extending Cayley's characterization of quartic symmetroids, by the property that the branch locus of the projection from a node consists of two cubic curves. This paper represents a first step towards the classification of all spectrahedra of a given degree and dimension.

The Grassmannian Gr(2, 5) is embedded in P 9 via the Plücker embedding. The intersection of two general PGL(10)-translates of Gr(2, 5) is a Calabi-Yau threefold X, and the intersection of the projective duals of the two translates is another Calabi-Yau threefold Y , deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain 'birational' Torelli statement for Calabi-Yau threefolds, namely, they are deformation equivalent, derived equivalent, and have isomorphic Hodge structures, but they are not birational.Proposition 1.1. Let g ∈ PGL(∧ 2 V ) be such that X g and Y g are of expected dimension. Then we have an equivalence of derived categories