Real $k$-flats tangent to quadrics in $\mathbb {R}^n$

Abstract:

Let $d_{k,n}$ and $\#_{k,n}$ denote the dimension and the degree of the Grassmannian $\mathbb {G}_{k,n}$, respectively. For each $1 \le k \le n-2$ there are $2^{d_{k,n}} \cdot \#_{k,n}$ (a priori complex) $k$-planes in $\mathbb {P}^n$ tangent to $d_{k,n}$ general quadratic hypersurfaces in $\mathbb {P}^n$. We show that this class of enumerative problems is fully real, i.e., for $1 \le k \le n-2$ there exists a configuration of $d_{k,n}$ real quadrics in (affine) real space $\mathbb {R}^n$ so that all the mutually tangent $k$-flats are real.