Computing β For Certain Schubert Subvarieties

Abstract

This thesis focuses on the invariant βY (L) attached to a triple (X, Y, L), where X is an irreducible projective variety, Y ⊂ X is a proper subscheme and L is a big or ample line bundle on X. This invariant has emerged independently in various papers, where it was used to attain new results in Diophantine geometry and mathematical physics. We pursue the task of computing βY (L) in the case where X = G(k, n), Y ⊂ X is a Schubert subvariety and L = OX (1). We develop a method by which we can compute βY (L) for any Schubert subvariety Y , provided that we satisfy some hypotheses; these hypotheses can be difficult to check, and the resulting formula can be difficult to evaluate. As our final result, we prove a concise formula for βY (L) when Y is a special type of Schubert subvariety, which we refer to as a maximal rectangle Schubert subvariety (abbreviated by MR Schubert subvariety).