A Depth Formula for Generic Singularities and Their Weak Normality

Let X be a smooth projective variety of dimension r and π:X → ?^m a generic projection with r + 1 ≤ m ≤ 2r. It is shown that, at any point on X′ = π(X) of multiplicity μ, off a closed subset of the triple locus of codimension four, the depth of the local ring is equal to r ? (μ ? 1)(m ? r ? 1). This leads to some improvements on the affirmation of a conjecture of Andreotti–Bombieri–Holm on the weak normality of X′ and a conjecture of Piene on the weak normality of Sing(X′).