Intersection Structures on Secants of Grassmannians

Abstract

Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. This thesis calculates dimensions of restricted, cyclic, and path geometric secants of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. Next, it defines a notion of expected dimension and gives a formula, which holds if the BDdG conjecture on non-defectivity of Grassmannians is true, for the dimension of all restricted secant varieties of Grassmannians. It also demonstrates example calculations in Macaulay 2 and points out ways to make these calculations more efficient. The thesis also shows a potential application to coding theory.