Linear subspaces, symbolic powers and Nagata type conjectures

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.

19 pages; made many minor improvements to exposition; one major improvement: replaced an example with 9 lines in P^4 by a family of examples with (n-1)^{n-2} lines in P^n for n >= 3

References in corpus (1)

• Star configurations in ${\mathbb{P}}^n$

Cited by in corpus (15)

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• Counting Lines on Quartic Surfaces
• Tropical approach to Nagata's conjecture in positive characteristic
• Local effectivity in projective spaces
• Inclics, galaxies, star configurations and Waldschmidt constants
• The Hilbert polynomial of a symbolic square
• On a conjecture of Demailly and new bounds on Waldschmidt constants in ${\mathbb{P}}^N$
• Asymptotic Hilbert Polynomial and a bound for Waldschmidt constants
• Containments of symbolic powers of ideals of generic points in $\PP^3$
• Postulation of generic lines and one double line in $\PP^n$ in view of generic lines and one multiple linear space
• Matroid configurations and symbolic powers of their ideals
• Actual and virtual dimension of codimension 2 general linear subspaces in ${\mathbb{P}}^n$
• Asymptotic invariants of generic initial ideals
• Asymptotic Hilbert Polynomial and limiting shapes
• On the fattening of ACM arrangements of codimension 2 subspaces in P^N