Abel maps and limit linear series

Definition 2.1. Fix integers d and r. A limit (linear) series on X of degree d and rank r is a collection consisting of an invertible sheaf L on X of degree d on Y and degree 0 on Z, and vector subspaces Vi ⊆ Γ(X,L) of dimension r + 1, for each i = 0, . . . , d, such that φ(Vi) ⊆ Vi+1 and φi(Vi+1) ⊆ Vi for each i. Given a limit series (L, V0, . . . , Vd), we denote by V Y,0 i the subspace of Vi of sections that vanish on Y , and by V Z,0 i the subspace of Vi of sections that vanish on Z. Also, let Vi|Y denote the subspace of Γ(Y,L|Y ) generated by Vi and Vi|Z that of Γ(Z,L|Z) generated by the same Vi. Of course, V Y,0 i is the kernel of the surjection Vi → Vi|Y , and V Z,0 i is the kernel of the surjection Vi → Vi|Z . Also, the map φ : Vi → Vi+1 has kernel V Z,0 i and image contained in V Y,0 i+1 , whereas φi : Vi+1 → Vi has kernel V Y,0 i+1 and image contained in V Z,0 i .