Jets of antimulticanonical bundles on Del Pezzo surfaces of degree ≤ 2

Abstract

In the paper under review, the authors study the jets of antimulticanonical bundles on certain del Pezzo surfaces. The main idea is to investigate the stratification of the surfaces given by the rank of the evaluation map with values in the second jet bundles of these antimulticanonical bundles. The second dual variety of the associated polarized surface is also considered. After reviewing some preliminaries in Section 1, the authors look at a del Pezzo surface S with K2S=2 in Section 2. Let L=−2KX. The rank of the evaluation map j2,x:H0(S,L)→(J2L)x is determined for every point x∈S. It follows that L is 2-jet spanned exactly on S−R, where R is the ramification divisor of the double cover S→P2 induced by |−KS|. Moreover, (−tKS) is 2-jet spanned for all t≥3. Section 3 and Section 4 are devoted to a del Pezzo surface S with K2S=1. This time, let L=−3KX. Again, the rank of j2,x is determined for every point x∈S. Furthermore, (−4KS) is 2-jet spanned exactly on S−Δ, where Δ is the set of singular points of the singular elements in the pencil |−KS|, and (−tKS) is 2-jet spanned for all t≥5. In Section 5, the authors apply the previous results to study the dual varieties Sv of the earlier polarized surfaces (S,L). It is proved that when K2S=2, the dual variety Sv is birational to S. When K2S=1, the dual variety Sv is a smooth rational curve parametrizing the pencil |−KS|.