Let ℌ be a class of n×n Hankel matrices HA whose entries, depending on a given matrix A, are linear forms in n variables with coefficients in a finite field 𝔽q. For every matrix in ℌ, it is shown that the varieties specified by the leading minors of orders from 1 to n-1 have the same number qn-1 of points in 𝔽qn. Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections.