Linear subspaces, symbolic powers and Nagata type conjectures

TL;DR

This paper develops asymptotic bounds for polynomial degrees vanishing on unions of disjoint r-dimensional planes in projective space, proposing that Nagata's conjecture for points is part of a broader pattern involving higher-dimensional subspaces.

Contribution

It introduces new asymptotic upper bounds and conjectures linking Nagata's point conjecture to a general phenomenon for higher-dimensional subspaces.

Findings

Established asymptotic upper bounds for degrees of forms vanishing on unions of planes

Proposed that Nagata's conjecture is a special case of a broader geometric pattern

Suggested new conjectures connecting subspace configurations to polynomial vanishing degrees

Abstract

Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in projective n space for n at least 2r+1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in the projective plane is not sporadic, but rather a special case of a more general phenomenon.