Restricted secant varieties of Grassmannians

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. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture [Baur et al. in Exp Math 16(2):239-250, 2007, Conjecture 4.1] on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay 2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.

Automated summary

Restricted secant varieties of Grassmannians are constructed by summing points corresponding to k-planes with the restriction of a prescribed intersection dimension. We study the dimensions of these restricted secant varieties and relate them to the dimensions of secants of Grassmannians using an incidence variety construction. We introduce the concept of expected dimension and provide a formula for the dimension of all restricted secant varieties of Grassmannians, assuming the non-defectivity conjecture on Grassmannians holds. We demonstrate example calculations in Macaulay 2 and suggest ways to improve computational efficiency. We also discuss a potential application to coding theory.