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Let $Q$ be the intersection of $k$ homogeneous quadrics (zero loci of homogeneous quadratic polynomials) restricted to $S^{n}\subset \mathbb{R}^{n+1}$. Consider the intersection of $Q$ with an hyperplane; it may dramatically depend on the choice of the hyperplane, but the question is the following: what can one say about the topology of the whole $Q$ once known something about the topology of the hyperplane section? Are there any well-known results, for instance about Betti numbers or other kind of "rough" informations?

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