Restricted Secant Varieties of Grassmannians

TL;DR

This paper investigates the dimensions of restricted secant varieties of Grassmannians, providing a formula under certain conjectural conditions, with computational examples and potential applications to coding theory.

Contribution

It introduces a dimension formula for restricted secant varieties of Grassmannians linked to a major conjecture, and explores computational methods and applications.

Findings

Derived a formula for the dimension of restricted secant varieties assuming the Baur-Draisma-deGraaf Conjecture.

Provided computational examples using Macaulay2.

Suggested applications to coding theory.

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. 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 Baur-Draisma-deGraaf Conjecture on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.