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The special Schubert calculus is real

Fulton asked: `How many solutions to a problem of enumerative geometry can be real?'. In this paper, we consider problems of enumerating p-planes having excess intersection with general linear subspaces and show that there is a choice of real linear subspaces osculating the rational normal curve so that all p-planes having excess intersection are real. This proves a special case of the conjecture of Shapiro and Shapiro.