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Logarithmic stable maps. (English) Zbl 1216.14023
Saito, Masa-Hiko (ed.) et al., New developments in algebraic geometry, integrable systems and mirror symmetry. Papers based on the conference “New developments in algebraic geometry, integrable systems and mirror symmetry”, Kyoto, Japan, January 7–11, 2008, and the workshop “Quantum cohomology and mirror symmetry”, Kobe, Japan, January 4–5, 2008. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-62-4/hbk). Advanced Studies in Pure Mathematics 59, 167-200 (2010).
Summary: We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semi-stable variety of form $$xy=0$$. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramified stable maps and stable relative maps. As an application, we obtain a modular desingularization of the main component of Kontsevich’s moduli space of elliptic stable maps to a projective space.
For the entire collection see [Zbl 1200.14002].

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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