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Modular compactifications of the space of pointed elliptic curves I

Part of:Curves

Published online by Cambridge University Press:  07 September 2010

David Ishii Smyth
David Ishii Smyth*
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: dsmyth@math.harvard.edu)
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Abstract

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We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack $\overline {\mathcal {M}}_{1,n}(m)$. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks $\overline {\mathcal {M}}_{1,\mathcal {A}}(m)$. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for $\overline {M}_{1,n}$.

Keywords

moduli of curveselliptic curvessingularities

MSC classification

Secondary:14H10: Families, moduli (algebraic)14H20: Singularities, local rings

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011, pp. 877 - 913
Copyright
Copyright © Foundation Compositio Mathematica 2010

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