Deformations of stable maps of curves

Given a smooth variety, X and a map g : X --> Delta such that g(-1) (0) is a normal crossing variety X-0 = X-1 boolean ORZ X-2, we consider stable maps F-0 : C-0 --> X-0 which appear as the central fibre of a family of maps [GRAPHIC] 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 boolean AND Z(i) in the Chow group A(0)(F-i(-1)(Z)). These conditions provide an elementary geometric justification for the work of Li and Ruan in [4] and of Gathmann in [2]. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.