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Real k-flats tangent to quadrics in R^n

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