Deformations of stable maps of curves

Abstract

Given a smooth variety, X and a map g: X→ Δ such that g −1(0) is a normal crossing variety X 0= X 1∪ Z X 2, we consider stable maps F 0: C 0→ X 0 which appear as the central fibre of a family of maps Splitting such a stable map up into F 1: C 1→ X 1 and F 2: C 2→ X 2, we derive conditions on the 0-cycle C i ∩ Z i in the Chow group A 0( F −1 i ( Z)). These conditions provide an elementary geometric justification for the work of Li and Ruan in [4] and of Gathmann in [2].