A degeneration of stable morphisms and relative stable morphisms
Abstract
The introduction is modified in the revised version. Also, many typos and errors were corrected. Let $W\to C$ be degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first constructed the stack of expanded degenerations of $W$. We then constructed the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to $W/C$. Using similar technique, for a pair of smooth variety and a smooth divisor $(Z,D)$, we constructed the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to $(Z,D)$. In the subsequent paper we will construct the relative GromovWitten invariants and prove a degeneration formula of GromovWitten invariants.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2000
 DOI:
 10.48550/arXiv.math/0009097
 arXiv:
 arXiv:math/0009097
 Bibcode:
 2000math......9097L
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Mathematical Physics;
 Mathematical Physics
 EPrint:
 48 pages