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Published in:
2017 | OriginalPaper | Chapter
On Singular Varieties with Smooth Subvarieties
Author : María del Rosario González-Dorrego
Published in: Singularities in Geometry, Topology, Foliations and Dynamics
Publisher: Springer International Publishing
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Let k be an algebraically closed field of characteristic 0. Let C be an irreducible nonsingular curve such that $${rC} = {S} \cap {F}, \ {r} \in \mathbb{N}$$ , where S and F are two surfaces and all the singularities of F are of the form z3 = xs − ys, s prime, with gcd(3, s) = 1. We prove that C can never pass through such kind of singularities of a surface, unless r = 3a, $${a} \ {\in} \ \mathbb{N}$$ . The case when the singularities of F are of the form z3 = x3s − y3s, $${s} \ {\in} \ \mathbb{N}$$ ; were studied in [3]. Next, we study multiplicity-r structures on varieties for any positive integer r. Let Z be a reduced irreducible nonsingular (n − 1)-dimensional variety such that $${rZ} = {X} \cap F$$ , where X is a normal n-fold with certain type of singularities, F is a (N − 1)-fold in $$\mathbb{P}^{N}$$ ; such that $${Z} \ \cap$$ Sing(X) ≠ ∅ We study the singularities of X through which Z passes.