Hypersurfaces containing involutive cones in projective symplectic spaces
Abstract
Let V be a complex symplectic vector space and let ℙ(V) carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety X⊂ℙ(V) which is a hypersurface of degree at least two in a hyperplane H is involutive if and only if it is a cone whose vertex contains the polar point σ(H). We give a short proof, construct the corresponding incidence spaces, compute their dimensions and express the incidence degrees as Segre-class integrals. In ℙ3 the incidence map is birational for every d≥ m≥ 2 except (m,d)=(2,2), and the same holds in ℙ5; the quadratic case is exceptional in every dimension, the incidence having generic degree 2n in ℙ2n−1. We also give a sufficient criterion for birationality in higher dimension, closed formulas in ℙ3 and coefficient formulas in ℙ5. For quadric cones in ℙ5 the degree is the product of shifted binomial factors and an irreducible polynomial of degree 36, and we explain where the shifted factors come from.
Related
- Algebraic hyperbolicity of very general hypersurfaces in projective spaces
- Relative symplectic cohomology in complex projective spaces
- Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs
- GKZ Systems and Gauss-Manin Connections of Families of Projective Hypersurfaces
- Retraction of the complement of smooth projective hypersurfaces to an n-dimensional Δ-complex
BibTeX
@misc{guedes2026hypersurfaces,
title = {{Hypersurfaces containing involutive cones in projective symplectic spaces}},
author = {Gabriel A. Guedes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19427},
eprint = {2609.19427},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19427},
}
RIS
TY - UNPB
TI - Hypersurfaces containing involutive cones in projective symplectic spaces
AU - Gabriel A. Guedes
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19427
UR - https://arxiv.org/abs/2609.19427
AB - Let V be a complex symplectic vector space and let ℙ(V) carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety X⊂ℙ(V) which is a hypersurface of degree at least two in a hyperplane H is involutive if and only if it is a cone whose vertex contains the polar point σ(H). We give a short proof, construct the corresponding incidence spaces, compute their dimensions and express the incidence degrees as Segre-class integrals. In ℙ^3 the incidence map is birational for every d≥ m≥ 2 except (m,d)=(2,2), and the same holds in ℙ^5; the quadratic case is exceptional in every dimension, the incidence having generic degree 2n in ℙ^2n−1. We also give a sufficient criterion for birationality in higher dimension, closed formulas in ℙ^3 and coefficient formulas in ℙ^5. For quadric cones in ℙ^5 the degree is the product of shifted binomial factors and an irreducible polynomial of degree 36, and we explain where the shifted factors come from.
ER -