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

### Antonio Lerario

SISSA, Trieste, Italy

## 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:

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:

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

## 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