Dicriticals of pencils and Dedekind’s Gauss lemma

The dicritical divisors of a pencil at a simple point of a surface constitute an important tool in affine algebraic geometry, i.e., in the study of polynomial rings. These dicriticals may be viewed as certain nodes of the singularity tree of a generic member of the pencil. Dedekind’s generalization of Gauss Lemma plays a significant role.