Singularity subschemes and generic projections

Abstract:

Corresponding to a morphism $f:V \to W$ of algebraic varieties (such that $\dim (V) \leqslant \dim (W)$), we construct a family of subschemes $S_1^{(q)}(f) \subset V$. When V and W are nonsingular, the $S_1^{(q)},q \geqslant 1$, induce a filtration of the set of closed points $x \in V$ such that the tangent space map $d{f_x}:T{(V)_x} \to T{(W)_{f(x)}}$ has rank $= \dim (V) - 1$. We prove that if V is a suitably embedded nonsingular projective variety and $\pi :V \to {{\mathbf {P}}^m}$ is a generic projection, then the $S_1^{(q)}(f)$ and certain fibre products of several of the $S_1^{(q)}(f)$ are either empty or smooth and of the smallest possible dimension, except in cases where $q + 1$ is divisible by the characteristic of the ground field. We apply this result to describe explicitly the ring homomorphisms ${\pi ^\ast }:{\hat {\mathcal {O}}_{{{\mathbf {P}}^m}\pi (x)}} \to {\hat {\mathcal {O}}_{V,x}}$ and (when $m \geqslant r + 1$) to study the local structure of the image $V’ = \pi (V) \subset {{\mathbf {P}}^m}$.