Reading about Bertini's theorem in a book of algebraic algebraic geometry, I see the statement:
Theorem: If $\Lambda$ is a linear system on a nonsingular variety X, then any general member $D \in \Lambda$, Sing$(D) \subset B_{\Lambda}$.
My question starts with how to see a divisor $D=m_1D_1+\cdots +m_kD_k$ as a subvariety of X. After all, singular points are defined in varieties.
My second question is whether the following statement may be a consequence of Bertini's theorem, but I don't know how to justify it:
Let $X \subset \mathbb{P}^n$ be a smooth irreducible projective variety. Let $H \subset \mathbb{P}^n$ be a hyperplane. The hyperplane section $H \cap X$ is smooth at $p$ if and only if the hyperplane $H$ is not tangent to $X$ at $p$.
Thanks