Effective cycles on blow-ups of Grassmannians

TL;DR

This paper investigates the structure of pseudoeffective cones on blow-ups of Grassmannians at points, identifying when Schubert classes generate these cones and describing the geometric implications of their failure to do so.

Contribution

It provides new bounds and descriptions for the pseudoeffective cones of blow-ups of Grassmannians, especially in cases where Schubert classes do not span the cones.

Findings

Schubert classes often span the pseudoeffective cones for small point sets

Sharp bounds are established for when Schubert classes fail to span

The geometric consequences of these failures are characterized

Abstract

In this paper we study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp bounds for when the Schubert classes fail to span and we describe the resulting geometry.