Jets of antimulticanonical bundles on Del Pezzo surfaces of degree \leq 2

Let S be a complex Del Pezzo surface with K_S^2 \leq 2, and let r=4-K_S^2. The line bundle L=-rK_S being very ample, we investigate the k-jet spannedness of -tK_S for t \geq r. A key point is the stratification given by the rank of the evaluation map j_2:S \times H^0(S,L) \to J_2L, with values in the second jet bundle of L, which puts in evidence some relevant loci related to both the intrinsic and the extrinsic geometry of S. In particular, the generic 2-jet spannedness of L allows us to consider the second dual variety of (S,L), parameterizing the osculating hyperplanes to S \subset P^6, embedded by |L|. Its behavior turns out to be completely different in the two cases K_S^2=2 and K_S^2=1.