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 X₀=X₁∪_ZX₂, we consider stable maps F₀:C₀→X₀ which appear as the central fibre of a family of maps Splitting such a stable map up into F₁:C₁→X₁ and F₂:C₂→X₂, we derive conditions on the 0-cycle C_i∩Z_i in the Chow group A⁰(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].