Abstract
Let X be a projective variety and let C be a rational normal curve on X. We compute the normal bundle of C in a general complete intersection of hypersurfaces of sufficiently large degree in X. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and normal bundles on each factor.
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