Let $X$ be a projective, normal variety over complex field with canonical singularities. Suppose $|D|$ is a basepoint free linear system, then is it true that the generic elements in $|D|$ are irreducible?


Besides, I noticed that something might related to "free linear system" (see Mori, Kollár "Birational geometry of algebraic varieties" Page 158, Lemma 5.17). Is "free linear system" the same as basepoint free linear system?



Irreducible divisor in a basepoint free linear system
No, the image of the associated morphism, $$\phi\_{|D|}:X\to \mathbb{P}^n,$$ might be a curve. If the image has dimension $\geq 2$, then the general member is irreducible. Look up "Bertini theorems".


