Advanced Studies in Pure Mathematics

Logarithmic stable maps

Bumsig Kim

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Abstract

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.

Article information

Source
New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), M. Saito, S. Hosono and K. Yoshioka, eds. (Tokyo: Mathematical Society of Japan, 2010), 167-200

Dates
Received: 12 September 2008
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543085924

Digital Object Identifier
doi:10.2969/aspm/05910167

Mathematical Reviews number (MathSciNet)
MR2683209

Zentralblatt MATH identifier
1216.14023

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Citation

Kim, Bumsig. Logarithmic stable maps. New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), 167--200, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05910167. https://projecteuclid.org/euclid.aspm/1543085924


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