Singularity Subschemes and Generic Projections

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