Irreducibility of universal hyperplane section In this question, Georges Elencwajg gives a fantastic geometric answer. However, this answer rests on the fact that the universal hyperplane section $\Omega_X$ is irreducible. I only know how to prove this by explicit computation (very nasty -- one splits into affine charts and rationally parameterizes each part) or by invoking a semicontinuity theorem in commutative algebra. Is there a slicker way to do this?

This is Theorem 5.8 of Harris's _Algebraic Geometry: A First Course_ (for arbitrary $X$). The proof given there is elementary.