Second Bertini theorem
In this post, I try to prove Bertini theorem partly, since there is no public reference about the case that the linear system is base point free. The proof below is just an analogue of the one that was provided by Hartshorne long ago in his book "Algebraic geometry". Theorem (second Bertini's theorem) . Let $X$ an algebraic variety over a closed field $k$ of characteristics $0$ and $dim X \geq 2$, let $\mathcal{L}$ be a linear system without fixed component on $V$, then for $D \in \mathcal{L}$ a general element of $\mathcal{L}$ we have $D_{sing} \subset X_{sing} \cup Bs \mathcal{L}$. Moreover, if $X$ is non-singular and a general element $D \in \mathcal{L}$ a big Cartier Divisor, then general elements of $|D|$ are non-singular and irreducible varieties. Proof : We will only prove the second statement. Since $D$ is a big Cartier divisor and $|D|$ a base point free linear system, then it could define a projective morphism $\phi_{D}: X \rightarrow \mathbb{P}(H^0(X...