For a very general hypersurface of degree d at least n+3 in P^{n+1}, any dominant rational map of prime degree to a smooth projective n-fold has a uniruled target, and the target is rationally connected when n is at most 3.
Vanishing cohomology on a double cover
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abstract
In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
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Morphisms from a very general hypersurface
For a very general hypersurface of degree d at least n+3 in P^{n+1}, any dominant rational map of prime degree to a smooth projective n-fold has a uniruled target, and the target is rationally connected when n is at most 3.