Abstract
Consider a generalized Grassmannian embedded in a projective space by a complete linear system of a positive generator of the Picard group. For a very general hypersurface
, we study subvarieties of
that are not of general type (or not of positive geometric genus). When the degree of the hypersurface is not small, we show that, under a certain condition on the parabolic subgroup P, such subvarieties are union of lines. Our result is a generalization of Clemens-Ran’s result concerning
.
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