Moduli of twisted spin curves

TL;DR

This paper introduces a new natural compactification of the stack of smooth r-spin curves called the stack of stable twisted r-spin curves, connecting it with twisted stable maps and simplifying various complex features.

Contribution

It provides a novel construction of the stack of stable twisted r-spin curves, identifying it with a special case of twisted stable maps and establishing isomorphisms with existing r-spin curve stacks.

Findings

Constructs a natural compactification of smooth r-spin curves.

Shows the stack of stable twisted r-spin curves is isomorphic to the existing r-spin curve stack.

Simplifies the understanding of complex features like torsion free sheaves and cohomology classes.

Abstract

In this note we give a new, natural construction of a compactification of the stack of smooth r-spin curves, which we call the stack of stable twisted $r$-spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G_m-spaces and Q-line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted r-spin curves to the stacks of stable r-spin curves, and show that they are isomorphisms. Many delicate features of r-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the d bar-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.