Polynomial Completion of Symplectic Jets and Surfaces Containing Involutive Lines

TL;DR

This paper investigates the polynomial completion of symplectic jets using advanced mathematical theories, establishing existence results, degree bounds, and counterexamples in specific dimensions related to involutive lines and hypersurfaces.

Contribution

It proves the existence of symplectic completions with degree bounds, disproves a conjectured bound in certain cases, and analyzes the rank of restriction maps for involutive lines in projective space.

Findings

Symplectic completions always exist using Andersén-Lempert Theory.

The degree bound conjectured by Dragt and Abell holds in physically relevant cases.

Counterexample: for 3-jets in dimension 4, the degree bound does not hold.

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 Anders\'en-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 $\iota:H^0(\mathbb P^3,\mathcal O(4))\rightarrow H^0(\Sigma,\mathcal O(4))$ has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if $(r,d)\neq (7,4)$ then the map $\iota$ has maximal rank.