Smooth projective varieties dominated by smooth quadric hypersurfaces in any characteristic (Q1346311)

We consider the following problem due to Remmert, Van de Ven and Lazarsfeld: Let \(X\) be a smooth projective variety defined over the complex number field \(\mathbb{C}\). Let \(f : G/P \to X\) be a surjective morphism where \(G\) is a simple classical group and \(P\) a maximal parabolic subgroup of \(G\). Then unless \(f\) is an isomorphism, \(X\) is isomorphic to a projective space. -- Here we show: Main theorem: Let \(X\) be a smooth projective variety, \(Q\) a smooth quadric hypersurface of dimension \(\geq 3\) and \(f\) a surjective morphism defined over an algebraically closed field of any characteristic (char \(k \neq 2)\). Assume that \(f\) is separable. Then unless \(f\) is an isomorphism, \(X\) is isomorphic to a projective space.