SUBSPACE ARRANGEMENTS OVER FINITE-FIELDS - COHOMOLOGICAL AND ENUMERATIVE ASPECTS

The enumeration of points on (or off) the union of some linear or affi ne subspaces over a finite field is dealt with in combinatorics via th e characteristic polynomial and in algebraic geometry via the zeta fun ction. We discuss the basic relations between these two points of view . Counting points is also related to the l-adic cohomology of the arra ngement (as a variety). We describe the eigenvalues of the Frobenius m ap acting on this cohomology, which corresponds to a finer decompositi on of the zeta function. The l-adic cohomology groups and their decomp osition into eigenspaces are shown to be fully determined by combinato rial data. Finally, it is shown that the zeta function is determined b y the topology of the corresponding complex variety in some important cases. (C) 1997 Academic Press.