Polynomial completion of symplectic jets and surfaces containing involutive lines

Abstract

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Andersén-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases. However, we disprove the degree bound for $3$-jets in dimension 4. This follows from the fact that if $\Sigma$ is the disjoint union of $r=7$ involutive lines in $\mathbb P^3$, then $\Sigma$ is contained in a degree $d=4$ hypersurface, i.e., the restriction morphism $\iota :H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(4))\rightarrow H^0(\Sigma ,\mathcal O_{\Sigma }(4))$ has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if $(r,d)\ne (7,4)$ then the map $\iota$ has maximal rank.