We classify those smooth (n-1)-folds in G(1,Pn) for which the restriction of the rank-(n-1) universal bundle has more than n+1 independent sections. As an application, we classify also those (n-1)-folds for which that bundle splits.
We prove that smooth subvarieties of codimension two in Grassmannians of lines of dimension at least six are rationally numerically subcanonical. We prove the same result for smooth quadrics of dimension at least six under some extra condition. The method is quite easy, and only uses Serre s construction, Porteous formula, and Hodge index theorem.
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