An algebraic proof of Zak’s inequality for the dimension of the Gauss image

A classical theorem in complex algebraic geometry states that, for any smooth projective variety, the Gauss map is finite; in particular, a smooth variety and its Gauss image have the same dimension (with the obvious exception of a linear space). Furthermore, even when the variety is not smooth, Zak proved a lower bound on the dimension of its Gauss image in terms of the dimension of its singular locus. Our purpose in this note is to reinterpret Gauss maps within a more general algebraic framework, and thus recover Zak’s bound on the dimension of the Gauss image as a special case of an interesting new bound on the analytic spread of a module of Kähler differentials. This connects that classical subject in complex geometry to recent research in commutative algebra concerning integral closures of modules. In particular, we give a new, purely algebraic proof of Zak’s theorem. We recall a precise version of Zak’s theorem. Let X be an irreducible projective variety of dimension d defined over an algebraically closed field k, considered with a fixed embedding X ⊂ P(V ) for some finite dimensional k-vector space V . The Gauss map is the rational map from X to the Grassmannian of projective d-planes in P(V ) assigning to each smooth k-point of X the projective tangent plane there,