Application of Bertini's theorem I am suffering with this rudimentary question related with Bertini's theorem. As one can see, in Hartshorne's book 'Algebraic Geomery V.1.2', there is one application of Bertini's theorem. I am really sorry, but I could not find the way how Bertini's theorem is used. Could any explain it with some details?
Thank you in advance.
 A: This answer is a bit verbose, since I don't know where your confusion lies. Hopefully it helps!
Let us first record the version of Bertini that Hartshorne wants to use:
Bertini's Theorem [Hartshorne II, 8.18, 8.18.1 and III, 7.9.1]. Let $Y$ be a subvariety of $\mathbf{P}^n_k$ where $k$ is an algebraically closed field, and where $Y$ has at most finitely many singular points. Then, the set of hyperplanes $H' \in \lvert H \rvert$ such that the hyperplane $H' \subseteq \mathbf{P}^n_k$ does not contain $Y$, and such that $H \cap Y$ is regular, forms an open dense subset of $\lvert H \rvert$. Moreover, if $\operatorname{dim} Y \ge 2$, then $H \cap Y$ is irreducible.
Now the statement you want to prove is:
Lemma. Let $C_1,\ldots,C_r$ be irreducible curves on a surface $X$, and let $D$ be a very ample divisor. Then almost all curves $D' \in \lvert D \rvert$ are irreducible, nonsingular, and meet each of the $C_i$ transversally.
Proof. Embed $X$ into some $\mathbf{P}^n_k$ using the complete linear system $\lvert D \rvert$. This means there is a surjection $\Gamma(\mathbf{P}^n_k,\mathcal{O}(1)) \twoheadrightarrow \Gamma(X,\mathscr{L}(D))$ by construction, and the corresponding rational map $\lvert H \rvert \dashrightarrow \lvert D \rvert$ given by $H' \mapsto H' \cap X$ is dominant.
We first apply Bertini's theorem to $Y = X$. Then, there exists an open dense subset $V$ of the complete linear system $\lvert H \rvert$ on $\mathbf{P}^n_k$ such that for all $H' \in V$, the intersection $D' = H' \cap X \in \lvert D \rvert$ is irreducible and nonsingular.
Next, we apply Bertini's theorem to $Y = C_i$ for each $i$. Then, there exist open dense sets $U_i$ of $\lvert H \rvert$ consisting of those hyperplanes $H'$ such that $C_i \cap H'$ is a proper closed subset of $C_i$ that consists of nonsingular points. This amounts to the fact that every point in $C_i \cap H'$ has multiplicity one, i.e., $C_i$ and $H'$ meet transversally.
Now the intersection $V \cap U_1 \cap \cdots \cap U_r$ of these open dense subsets in $\lvert H \rvert$ is also open and dense. The image of this open dense set in $\lvert D \rvert$ via the map $\lvert H \rvert \dashrightarrow \lvert D \rvert$ is also dense, and this set in $\lvert D \rvert$ consists of those $D'$ as in the statement of the Lemma. $\blacksquare$
