JournalsjemsVol. 18, No. 2pp. 353–379

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

  • Antonio Lerario

    SISSA, Trieste, Italy
Complexity of intersections of real quadrics and topology of symmetric determinantal varieties cover
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Abstract

Let WW be a linear system of quadrics on the real projective space RPn\mathbb R P^n and XX be the base locus of that system (i.e. the common zero set of the quadrics in WW). We prove a formula relating the topology of XX to the one of the discriminant locus ΣW\Sigma_W (i.e. the set of singular quadrics in WW). The set ΣW\Sigma_W equals the intersection of WW with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of WW) and we set {ΣW(r)}r1\{\Sigma_W^{(r)}\}_{r\geq 1} for its singular point stratification, i.e. ΣW(1)=ΣW\Sigma_W^{(1)}=\Sigma_W and ΣW(r)=Sing(ΣW(r1))\Sigma_W^{(r)}=\textrm{Sing}\big( \Sigma_W^{(r-1)}\big). With this notation, for a generic WW the mentioned formula writes:

b(X)b(RPn)+r1b(PΣW(r)).b(X) \leq b(\mathbb R P^n)+ \sum_{r \geq 1}b(\mathbb{P}\Sigma_W^{(r)}).

In the general case a similar formula holds, but we have to replace each b(PΣW(r))b(\mathbb{P}\Sigma_W^{(r)}) with 12b(Σϵ(r))\frac{1}{2}b(\Sigma_\epsilon^{(r)}), where Σϵ\Sigma_\epsilon equals the intersection of the discriminant hypersurface with the unit sphere on the translation of WW in the direction of a small negative definite form. Each Σϵ(r)\Sigma_\epsilon^{(r)} is a determinantal variety on the sphere Sk1S^{k-1} defined by equations of degree at most n+1n+1 (here kk denotes the dimension of WW); we refine Milnor's bound, proving that for such affine varieties b(Σϵ(r))O(n)k1b(\Sigma_\epsilon^{(r)})\leq O(n)^{k-1}.

Since the sum in the above formulas contains at most O(k)1/2O(k)^{1/2} terms, as a corollary we prove that if XX is any intersection of kk quadrics in RPn\mathbb R P^n then the following sharp estimate holds:

b(X)O(n)k1.b(X) \leq O(n)^{k-1}.

This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape O(n)2k+2O(n)^{2k+2}).

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. 353–379

DOI 10.4171/JEMS/592