When Ramification points meet

Summary

This paper contains three applications of the technique of limit series (our [1986]) to the theory of ramification of linear series on smooth curves, and curves of compact type, overC.

Let {L _t ‖t|<ε}, be a family of linear series on a smooth family of smooth curves {C _t }, and letp ₁(t),p ₂(t)∈C _t be sections of the family which coincide (only) att=0. Setp=p ₁())=p ₂(0)∈C ₀.

We first give a condition related to the Schubert calculus which must be satisfied by the ramification series$\alpha ^{L_0 } (p)$ and the$\alpha ^{L_t } (p_i (t))$. We then take up the converse problem: In what ways can a given ramification point arise as a limit? We show that if the ramification point isdimensionally proper in the sense of our [1986], then families of every kind allowed by the Schubert calculus condition can actually be constructed. Finally, we prove that dimensional propriety is in a strong sense an open condition, so that ramification points constructed as above are again dimensionally proper.

In the body of the paper we work not with pairs of points, as above, but with arbitrary finite collections of points approaching (possibly) several points of the limit curve. Further, by their nature, the results are valid for families of curves of compact type.