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Published in:
2017 | OriginalPaper | Chapter
Singular Intersections of Quadrics I
Author : Santiago López de Medrano
Published in: Singularities in Geometry, Topology, Foliations and Dynamics
Publisher: Springer International Publishing
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Let $${Z} \ \subset \ \mathbb{R}^{n} $$ be given by k + 1 equations of the form $$\sum\limits_{i=1}^{n} {A}_{i}{{x}_{i}^{2}} = {0}, \qquad\qquad \sum\limits_{i=1}^{n} {{x}_{i}^{2}} = 1,$$ where $${A}_{i} {\in} \mathbb{R}^{k}$$ . It is well known that the condition for Z to be a smooth variety (known as weak hyperbolicity) is that the origin in $$\mathbb{R}^{k}$$ is not a convex combination of any collection of k of the vectors A i . We interpret this condition as a transversality property in order to approach the case when it is singular and we extend some results known for the smooth case, in particular the computation of the homology groups of Z in terms of the combinatorics of the natural quotient polytope. We show that Z cannot be an exotic homotopy sphere nor a non-simply connected homology sphere and use this to show that, except for some clearly characterized degenerate cases, when Z is not smooth it cannot be a topological or even a homological manifold.