A refinement of the strange duality conjecture on K3 surfaces

Abstract: Let $v,w$ be primitive Mukai vectors of positive rank on a polarized K3 surface with $χ(v\otimes w)=0$. Marian and Oprea conjectured that the strange duality map $$H^{0(M(v),Θ_w)^\vee\to} H<sup>0(M(w),Θ_v)$$ is either zero or an isomorphism. We give three families of counterexamples to this conjecture. In each of them the map is nonzero but fails to be an isomorphism: in the first (a quartic surface) it is not surjective; in the second (Picard rank one, including very general polarized K3 surfaces) the source has strictly larger dimension than the target; and in the third (an elliptic surface) source and target both have dimension six, yet the map has rank five. The three examples share a common mechanism: one of the moduli spaces carries an effective divisor of square $-2$ on which a theta line bundle has negative degree, so that this theta line bundle is not nef and has nonvanishing higher cohomology, and the theta sections produced by the strange duality map vanish to excess order along this divisor. We therefore propose a refined conjecture, in which both theta line bundles are assumed to be big and nef, for instance ample.