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

Abstract

Let $W$ be a linear system of quadrics on the real projective space $\mathbb R P^n$ 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 $\Sigma\_W$ (i.e. the set of singular quadrics in $W$). The set $\Sigma\_W$ 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 ${\Sigma\_W^{(r)}}\_{r\geq 1}$ for its singular point stratification, i.e. $\Sigma\_W^{(1)}=\Sigma\_W$ and $\Sigma\_W^{(r)}=\textrm{Sing}\big( \Sigma\_W^{(r-1)}\big)$. With this notation, for a generic $W$ the mentioned formula writes: $$ b(X) \leq b(\mathbb R P^n)+ \sum\_{r \geq 1}b(\mathbb{P}\Sigma\_W^{(r)}). $$ In the general case a similar formula holds, but we have to replace each $b(\mathbb{P}\Sigma\_W^{(r)})$ with $\frac{1}{2}b(\Sigma\_\epsilon^{(r)})$, where $\Sigma\_\epsilon$ equals the intersection of the discriminant hypersurface with the unit sphere on the translation of $W$ in the direction of a small negative definite form. Each $\Sigma\_\epsilon^{(r)}$ is a determinantal variety on the sphere $S^{k-1}$ defined by equations of degree at most $n+1$ (here $k$ denotes the dimension of $W$); we refine Milnor's bound, proving that for such affine varieties $b(\Sigma\_\epsilon^{(r)})\leq O(n)^{k-1}$. Since the sum in the above formulas contains at most $O(k)^{1/2}$ terms, as a corollary we prove that if $X$ is any intersection of $k$ quadrics in $\mathbb R P^n$ then the following sharp estimate holds: $$ b(X) \leq O(n)^{k-1}. $$ This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape $O(n)^{2k+2}$).