abstract: A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. I will revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. When the spaces in question are either smooth, quasi-projective varieties, or closed three-dimensional manifolds, the respective jump loci are much more tightly connected, albeit in different ways.