Alternate compactifications of the moduli space of genus one maps

Details Alternate compactifications of the moduli space of genus one maps Alternate compactifications of the moduli space of genus one maps

2010-05-09

arxiv.org/abs/1005.1431v1

Mathematics

Algebraic Geometry

38 pages, 4 figures

Scientific paper

We extend the definition of an m-stable curve introduced by Smyth to the setting of maps to a projective variety X, generalizing the definition of a Kontsevich stable map in genus one. We prove that the moduli problem of n-pointed m-stable genus one maps of class \beta is representable by a proper Deligne-Mumford stack \sMbar_{1,n}^m(X, \beta ) over Spec Z[1/6]. For X = P^r, we explicitly describe all of the irreducible components of \sMbar_{1,n}(P^r,d) and \sMbar_{1,n}^m(P^r,d), and in particular deduce that \sMbar_{1,n}^m(P^r,d) is irreducible for m >= min(r,d) + n. We show that \sMbar_{1,n}^m(P^r,d) is smooth if d+n <= m <= 5.

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