Actual and virtual dimension of codimension 2 general linear subspaces in $\mathbb{P}^n$

TL;DR

This paper calculates the expected dimension of hypersurfaces containing general codimension 2 linear subspaces in projective space and discovers unexpected hypersurfaces using Veneroni maps, extending previous examples.

Contribution

It introduces a method to compute the virtual dimension of such hypersurfaces and identifies new unexpected hypersurfaces via Veneroni maps, expanding prior work.

Findings

Computed virtual dimensions for hypersurfaces with linear subspaces

Identified families of unexpected hypersurfaces

Extended previous examples with rigorous proofs

Abstract

In the paper we compute the virtual dimension (defined by the Hilbert polynomial) of a space of hypersurfaces of given degree containing $s$ codimension 2 general linear subspaces in $\mathbb{P}^n$. We use Veneroni maps to find a family of unexpected hypersurfaces (in the style of B. Harbourne, J. Migliore, U. Nagel, Z. Teitler) and rigorously prove and extend examples presented in the paper by B. Harbourne, J. Migliore and H. Tutaj-Gasi\'nska.