Alternate compactifications of the moduli space of genus one maps |
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Authors: | Michael Viscardi |
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Institution: | 1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA
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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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