Abstract
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 β is representable by a proper Deligne–Mumford stack \({\overline{\mathcal {M}}_{1,n}^{m}(X,\beta)}\) over Spec \({\mathbb {Z}[1/6]}\) . For \({X=\mathbb {P}^{r},}\) we show that \({\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^{r},d)}\) is irreducible for m sufficiently large. We also show that \({\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^r,d)}\) is smooth if d + n ≤ m ≤ 5.
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Viscardi, M. Alternate compactifications of the moduli space of genus one maps. manuscripta math. 139, 201–236 (2012). https://doi.org/10.1007/s00229-011-0513-2
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DOI: https://doi.org/10.1007/s00229-011-0513-2