T. Szemberg: Linear subspaces and symbolic powers

Abstract

It is a report on a work in progress with
Brian Harbourne, Marcin Dumnicki,
Joaquim Roe and Halszka Tutaj-Gasinska.

Let $L_1,...,L_s$ be r-dimensional linear subspaces of
the projective space $\mathbb{P}^n$. Let $ J=J_{n,r,s}$ be the
homogeneous ideal of the union of these subspaces.
One of classical problems in algebraic geometry
and in the commutative algebra is to estimate
the initial degree of J, i.e. the least degree of
a non-zero form vanishing along all subspaces $L_i$.
If r=0, i.e. if we deal with points, then this is
very easy. Already for lines the problem has no
easy solution but the solution is known thanks to
works of Hartshorne and Hirschowitz in the 80's.
For higher dimensional subspaces nothing is known.
We study an asymptotic variant of this question
and exhibit a tower of differential equations
hidden behind.