Fubini's theorem in codimension two
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abstract
We classify codimension two complex submanifolds of projective space X n having the property that any line through a general point x X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarre-de Jong conjecture and of varieties whose Fano variety of lines has dimension 2n - 4. Walter de Gruyter Berlin New York 2009.