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.[1]

In modern terms, it is a subsystem of the linear system associated to the normal bundle to $C\hookrightarrow Y$ . Note a characteristic system need not 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.

References

Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip (2011). Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften. Vol. II, with a contribution by Joseph Daniel Harris. Heidelberg: Springer. p. 3. doi:10.1007/978-1-4757-5323-3. MR 2807457.

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[1] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip (2011). Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften. Vol. II, with a contribution by Joseph Daniel Harris. Heidelberg: Springer. p. 3. doi:10.1007/978-1-4757-5323-3. MR 2807457.