Moduli of twisted spin curves

By D. Abramovich, Tyler Jarvis. 2001

104 citations recorded by the source corpus.

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 ∂-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7c46339a2ac5025dd3c6640dbcab44fa6197ffc

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In Geometry and Topology

1. Ingrid Bauer
2. G. Farkas
3. Lucia Caporaso
4. F. Catanese
5. A. Chiodo
6. D. Abramovich
7. A. Verra
8. Angelo Vistoli

In Mathematical Physics

1. G. Farkas
2. Ingrid Bauer
3. A. Chiodo
4. F. Catanese
5. A. Verra
6. Lucia Caporaso
7. D. Abramovich
8. Angelo Vistoli

Citations in the corpus

• Twisted Bundles and Admissible Covers (2001, background): “A somewhat less detailed version of this argument can be found in Abramorvch and Jarvis ( [2003] ).”
• Towards an enumerative geometry of the moduli space of twisted curves and rth roots (2006, background): “ell known that one can construct a proper and smooth stack M¯r g,n of rth roots of the canonical bundle of genus-gcurves or of any of its powers and twists at the markings x 1,x 2,...,xn, see [Ja00], [AJ03], [CCC], and [Ch2]. In Theorem 1.1.2, we determine the Chern character of ch(R•π∗S), where π: ”
• The geometry of the moduli space of odd spin curves (2010, background): “ gives rise to a finite covering π : S − g →Mg. Furthermore,S − g has a modular meaning being isomorphic tothecoarse moduli space of theDeligne-Mumford stack of odd stable spin curves, cf. [C], [CCC], [AJ]. The map π is branched along the boundary of Mg and one expects K S− g to enjoy better positivi”
• The birational type of the moduli space of even spin curves (2008, methodology): “A compactification Sg of Sg over Mg is obtained by considering the coarse moduli space of the stack of stable spin curves of genus g (cf. [C], [CCC] and [ AJ ]).”
• Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “For a subtle example see [Jar00, Definition 2.5] as well as [ AJ03 , §4.2-4.4], where an equivalence analogous to our Theorem 5.3.6 below is given.”
• Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations (2008, unknown): “1.1 (for the A-singularity case see [ 2 , 5, 25, 26, 35, 36]).”
• Moduli of roots of line bundles on curves (2004, background): “We then prove that, for r ≥ 3, the previously constructed moduli spaces of limit roots, which we denote by S r, l g , as well as those constructed in [14], [15], [1] and [7], do not embed in the universal Picard scheme.”