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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Singularity subschemes and generic projections

Author: Joel Roberts
Journal: Trans. Amer. Math. Soc. 212 (1975), 229-268
MSC: Primary 14E25; Secondary 14N05, 14B05
MathSciNet review: 0422274
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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}$.

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Keywords: Projective algebraic variety, generic projection, Kähler differentials, algebra of principal parts, noetherian scheme, scheme-theoretic fibre, singularity subscheme
Article copyright: © Copyright 1975 American Mathematical Society