Real 𝑘-flats tangent to quadrics in ℝⁿ

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