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 $C\hookrightarrow Y$. 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.

References

• Enrico Arbarello, Maurizio Cornalba, and Phillip Griffiths, Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, vol. 268, Springer, Heidelberg, 2011. MR2807457 doi:10.1007/978-1-4757-5323-3