On the Zeta Function of a Hypersurface: IV. A Deformation Theory for Singular Hypersurfaces

Abstract

-may be described in terms of an alternating product [3, Th. 20.1], the most important term of which comes from S, the subspace of R consisting of elements satisfying certain growth conditions. On the other hand if f is defined over an algebraic number field then for almost all primes, S R. In [2, ? 5] we considered the variation of St as f ran through a rationally parametrized family of forms whose generic element represents (in characteristic zero) a non-singular hypersurface in general position. In the present article we extend this theory to the case of families of hypersurfaces whose generic elements may be singular. We show that many of the preceding results remain valid and, in particular, that St has a basis depending rationally on the parameters, and that this basis has essential singularities only at parameter values for which the dimension of S is less than the generic dimension and that St St for all f in some Zariski open set (which may be empty for some primes). We find that the variation of the endomorphism a* of M is again given by a solution matrix of a system of linear differential equations (Equation 6.5 below). It is known from the work of Katz [4], that in the non-singular case, these differential equations are (with very minor modifications) the same as the classical differential equations satisfied by the elements of the nth deRham space (holomorphic n forms modulo exact ones) of the complement H' in projective n-space (characteristic zero) of the algebraic set defined by