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 $\{{\Sigma }_W^{(r)}{\}}_{r\ge 1}$ for its singular point stratification, i.e. ${\Sigma }_W^{(1)}={\Sigma }_W$ and ${\Sigma }_W^{(r)}=\text{Sing}\left({\Sigma }_W^{(r-1)}\right)$. With this notation, for a generic $W$ the mentioned formula writes:

In the general case a similar formula holds, but we have to replace each $b(\mathbb{P}{\Sigma }_W^{(r)})$ with $\frac{1}{2}b({\Sigma }_{ϵ}^{(r)})$, where ${\Sigma }_{ϵ}$ 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 ${\Sigma }_{ϵ}^{(r)}$ 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 $b({\Sigma }_{ϵ}^{(r)})\le O(n{)}^{k-1}$.

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: $b(X)\le O(n{)}^{k-1}.$ This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape $O(n{)}^{2k+2}$).