Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems

Abstract

It is a fundamental challenge for many problems of significant current interest in algebraic geometry and commutative algebra to understand symbolic powers $I^{(m)}$ of homogeneous ideals $I$ in polynomial rings, particularly ideals of linear varieties. Such problems include computing Waring ranks of polynomials, determining the occurrence of equality $I^{(m)} = I^m$ (or, more generally, of containments $I^{(m)} \subseteq I^r$), computing Waldschmidt constants (i.e., determining the limit of the ratios of the least degree of an element in $I^{(m)}$ to the least degree of an element of $I^m$), and studying major conjectures such as Nagata’s Conjecture and the uniform SHGH Conjecture (which respectively specify the Waldschmidt constant of ideals of generic points in the plane and the Hilbert functions of their symbolic powers).