Alternate compactifications of the moduli space of genus one maps
Tóm tắt
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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