Geodesic
Complexity of intersections of real quadrics and topology of symmetric determinantal varieties
Journal of the European Mathematical Society, Tome 18 (2016) no. 2, pp. 353-379

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Let W be a linear system of quadrics on the real projective space RPn and X be the base locus of that system (i.e. the common zero set of the quadrics in W). We prove a formula relating the topology of X to the one of the discriminant locus ΣW​ (i.e. the set of singular quadrics in W). The set ΣW​ equals the intersection of W with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of W) and we set {ΣW(r)​}r≥1​ for its singular point stratification, i.e. ΣW(1)​=ΣW​ and ΣW(r)​=Sing(ΣW(r−1)​). With this notation, for a generic W the mentioned formula writes:
DOI : 10.4171/jems/592
Classification : 14-XX, 55-XX
Keywords: Real algebraic geometry, real quadrics, homological compexity, determinantal varieties 
Antonio Lerario. Complexity of intersections of real quadrics and topology of symmetric determinantal varieties. Journal of the European Mathematical Society, Tome 18 (2016) no. 2, pp. 353-379. doi: 10.4171/jems/592
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     author = {Antonio Lerario},
     title = {Complexity of intersections of real quadrics and topology of symmetric determinantal varieties},
     journal = {Journal of the European Mathematical Society},
     pages = {353--379},
     year = {2016},
     volume = {18},
     number = {2},
     doi = {10.4171/jems/592},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/592/}
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