Let $S \subset \mathbb{P}^3$ be a surface different to a cone. If $S$ is a smooth variety, then its generic hyperplane section is irreducible. I can imagine a similar result holds for a surface with isolated singularities....
Are there necessary and sufficient conditions over $S$ that guarantee an irreducible hyperplane section?
I understand my question may related to the Lefschetz hyperplane theorem..However, its singular version seems hard, and I cannot find a version for not normal varieties.