Some p - ranks Related to Finite Geometric Struc

The prank of the point-hyperplane incidence matrix A of PG(n; p e) is well-known. Let A S be the submatrix formed by the rows of A indexed by an arbitrary subset S of the points. We show that the prank of A S is related to the Hilbert function (or a modiication thereof) for I(S), the ideal of FX 0 ; X 1 ; : : :; X n ] generated by all homogeneous polynomials vanishing on S. This leads to a determination of rank p (A S) in case S is a naturally embedded Grassmann variety. The cases when S is a quadric or a Hermitian variety have been treated by Blokhuis and the author 2] and the author 10] respectively. 1 HILBERT FUNCTIONS AND pRANKS Let F = GF(q), q = p e , and let A be the incidence matrix of points versus hyperplanes of PG(n; F). Thus A is a square matrix of size N = (q n+1 ?1)=(q ?1) having entries 1 and 0 corresponding to incident and non-incident point-hyperplane pairs. Now let A S be an s N submatrix of A, whose rows are indexed by an s-subset S of the points of PG(n; F). The intent of Theorem 1 is to describe a general approach to nding the prank of A S. This approach makes use of a modiication of the Hilbert function of I(S), the ideal in the polynomial ring R := FX 0 ; X 1 ; : : :; X n ] generated by all homogeneous polynomials which vanish on S. Much of our terminology and background results are standard in algebraic geometry; see eg. 7].