Finite generation and the Gauss process

Convergence of the Gauss resolution process for a complex singular foliation of dimension r is shown to be equivalent to finite type of a graded sheaf which is built using base (r+2) expansions of integers. As applications it is calculated which foliations coming from split semisimple representations of commutative Lie algebras can be resolved torically with respect to an eigenspace decomposition and it is shown that Gaussian resolutions stabilize for irreducible normal projective varieties with foliations of dimension r for which (r+1)H+K is finitely generated with Iitaka dimension less than two where H is a hyperplane section and K a canonical divisor of the foliation. For normal irreducible complex projective varieties with very ample divisor H and a resolvable foliation, functorial locally closed conditions on vector subspaces X \subset |iH| are given which hold for large i and ensure that blowing up the base locus of X and one further Gaussian blowup resolves the foliation.