Qualifying Examination
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so ‖f‖p ≤ ‖f‖q · μ(X) 1 p − 1 q . Hence if f is in Lq, the left-hand side is finite hence so is the right-hand side, so f is in Lp. Also, the inequality shows that if ‖f‖p is small then ‖f‖q is also small, hence the inclusion Lq ↪→ Lp is continuous 2. Let X ⊂ Pn be an irreducible projective variety of dimension k, G(`, n) the Grassmannian of `-planes in Pn for some ` < n− k, and C(X) ⊂ G(`, n) the variety of `-planes meeting X. Prove that C(X) is irreducible, and find its dimension. Solution. We have the diagram
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