Characteristic linear system of an algebraic family of curves

In algebraic geometry, the characteristic linear system of a family of curves on an algebraic surface "Y" for a curve "C" in the family is a linear system formed by the curves in the family that are infinitely near "C".
In modern terms, it is a subsystem of the linear system associated to the normal bundle to formula_1. Note a characteristic system need to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness.