I apologize in advance if this question is too elementary. Let $f\colon X\to Y$ be a flat proper morphism of algebraic varieties over an algebraically closed field, and assume that $Y$ is irreducible (not all my assumptions might be necessary). Let $y\in Y$ be a closed point such that the fiber $f^{-1}(y)$ is a smooth irreducible variety.

**Questions.** 1) Is it true that the fiber over the generic point of $Y$ is irreducible and regular?

2) Is it true that the set of points of $Y$, whose fibers are irreducible and regular, is Zariski open?