Exact birational range in higher-dimensional projective symplectic spaces

Determine the exact birational range of the involutive cone incidence projection for hyperplane-supported involutive cones of degree m contained in projective symplectic spaces \(\mathbb{P}^{2n-1}\) with \(n\ge4\), beyond the sufficient birationality criteria established in the paper.

Background

The paper studies the incidence projection from the parameter space of degree-$m$ hypersurface cones whose vertex contains the symplectic polar point of the supporting hyperplane to the locus of degree-$d$ hypersurfaces containing such cones. Birationality is proved completely in $\mathbb{P}^3$ and $\mathbb{P}^5$, while a sufficient criterion is established in higher odd dimensions.

For $n\ge4$, the sufficient criterion does not determine all cases. The unresolved issue is to identify precisely when a general hypersurface in the incidence image contains a unique involutive cone and when the incidence projection has higher generic degree.