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Restricted secant varieties of Grassmannians

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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 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.

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Acknowledgements

Oeding thanks Roland Abauf, Elisa Postinghel, for initial discussions on this topic. Bidleman thanks Matt Speck and Colby Muir for discussions on the subject.

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  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

    Dalton Bidleman & Luke Oeding

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  1. Dalton Bidleman
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Correspondence to Luke Oeding.

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Bidleman, D., Oeding, L. Restricted secant varieties of Grassmannians. Collect. Math. 75, 545–565 (2024). https://doi.org/10.1007/s13348-023-00399-4

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