Let be a linear system of quadrics on the real projective space and be the base locus of that system (i.e. the common zero set of the quadrics in ). We prove a formula relating the topology of to the one of the discriminant locus (i.e. the set of singular quadrics in ). The set equals the intersection of with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of ) and we set for its singular point stratification, i.e. and . With this notation, for a generic the mentioned formula writes:
In the general case a similar formula holds, but we have to replace each with , where equals the intersection of the discriminant hypersurface with the unit sphere on the translation of in the direction of a small negative definite form. Each is a determinantal variety on the sphere defined by equations of degree at most (here denotes the dimension of ); we refine Milnor's bound, proving that for such affine varieties .
Since the sum in the above formulas contains at most terms, as a corollary we prove that if is any intersection of quadrics in then the following sharp estimate holds:
This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape ).
Cite this article
Antonio Lerario, Complexity of intersections of real quadrics and topology of symmetric determinantal varieties. J. Eur. Math. Soc. 18 (2016), no. 2, pp. 353–379DOI 10.4171/JEMS/592