Alternate compactifications of the moduli space of genus one maps

TL;DR

This paper extends the concept of m-stable curves to genus one maps, proving the moduli space is a proper Deligne-Mumford stack, and describes its components and smoothness properties for projective space targets.

Contribution

It generalizes Smyth's m-stability to genus one maps and explicitly characterizes the irreducible components of their moduli spaces for projective targets.

Findings

Moduli space of genus one maps is a proper Deligne-Mumford stack.

Irreducibility of the moduli space for certain m values.

Conditions for smoothness of the moduli space.

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 \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.