Let $S$ be a complex non-singular projective surface embedded in some $\mathbb P^n$. Thanks to the Bertini's theorem (Hartshorne theorem II.8.18) there exists a hyperplane $H\subseteq\mathbb P^n$ such that $H\cap S$ is a smooth projective curve (so irreducible) and moreover the generic hyperplane $L$ in the linear system $|H|$ has such a property.
Now suppose that $H'\in |H|$ is a hyperplane which doesn't satisfy the Bertini theorem, what can we say about the "curve" $H'\cap S$? Is it reduced, connected or irreducible? Moreover what about the singular points of $H'\cap S$? They are nodes?
Practically I'd like to know some information about the "bad" hyperplane sections of $S$.
The degree $d$ curves in $\mathbb P^2$ form a linear system parameterized by $\mathbb P^{(d+3)d/2}$. If we embed $\mathbb P^2$ into $\mathbb P^N$ via the $d$-uple embedding, these are the linear system of hyperplane sections.
Now what you can say about the structure of singular degree $d$ curves?