We associate a tangent lattice to a primitive integer lattice and study its typical shape. This is motivated by Peyre’s program on the freeness of rational points on Fano varieties: A primitive integer lattice can be regarded a point on a Grassmanian, and the shape of its tangent lattice determines this point’s freeness.

The reason behind this interest in freeness is Manin’s conjecture about the number of rational points of bounded height on Fano varieties: This number might be dominated by “bad” points on subvarieties, or more generally, a thin set of “bad“ points that has to be excluded in the count. Peyre proposed to exclude points of low freeness, so that points of high freeness should conform to the asymptotic formula proposed by Manin’s conjecture and its variants. Our analysis verifies this for Grassmanians by proving that there are relatively few points of low freeness.

This is joint work with Tim Browning and Tal Horesh.

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