Mathematics – Algebraic Geometry
Scientific paper
2010-05-09
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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