Variation of Ramification Loci of Generic Projections

Let X C P N = P   N Kbe a subvariety of dimension n and ∧ P N be a generic linear subspace of dimension N k ‐ 1 with k ≥ n. Then the linear projection π∧: X → P k is a finite map. Let R (π∧) be its ramification locus. In this paper we study the map from the Grassmannian G(N ‐ k ‐ 1, N ) of planes of dimension N ‐ k ‐ 1 in P N to the Hilbert moduli space given by ∧ R (π∧). We wish to compute in particular the dimension, say, n of the image of this map. The motivation of this question comes from the fact that these ramification cycles are closely related to the Stückrad ‐ Vogel cycle. We show that n is just the transcendence degree of a certain part of this cycle. The main result is that, under some mild hypothesis, in case of a projection X → P n , i.e., k = n, the map ∧ → R (π∧) is generically finite and so n takes its maximal possible value. Moreover, we show that in the case of smooth surfaces X ∇ P 4 and generic projections onto P 3 this map is again generically finite if the normal bundle of X in P 4 is sufficiently positive.