Abstract |
Classical Schubert calculus deals with the intersection of Schubert varieties in general position. We present an attempt at developing such a theory over the reals. By the title we understand the investigation of the expected number of points of intersection of real Schubert varieties in random position. We define a notion of expected degree of real Grassmannians that turns out to be the key quantity governing questions of random incidence geometry. Using integral geometry, we prove a result that decouples a random incidence geometry problem into a volume computation in real projective space and the determination of the expected degree. Over the complex numbers, the same decoupling result is a consequence of the ring structure of the cohomology of the Grassmannian. We prove an asymptotically sharp upper bound on the expected degree of the real Grassmannians G(k,n). Moreover, if both k and n go to infinity, the expected degree turns out to have the same asymptotic growth (in the logarithmic scale) as the square root of the degree of the corresponding complex Grassmannian. This finding is in the spirit of the real average Bezout's theorem due to Shub and Smale. In the case of the Grassmannian of lines, we can provide a finer asymptotic. |