Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extended clutching construction for the moduli of stable curves

Alexander Polishchuk

Abstract

We give a description of the formal neighborhoods of the components of the boundary divisor in the Deligne-Mumford moduli stack $\overline{\mathcal M}_g$ of stable curves in terms of the extended clutching construction that we define. This construction can be viewed as a formal version of the analytic plumbing construction. The advantage of our formal construction is that we can control the effect of changing formal parameters at the marked points that are being glued. As an application, we prove that the infinitesimal neighborhood of the boundary component $Δ_{1,1}$ in $\overline{\mathcal M}_2$ is canonically isomorphic to the infinitesimal neighborhood of the zero section in the normal bundle. As another application, we show how to study the period map near the boundary components $Δ_{g_1,g_2}$ in terms of the coordinates coming from our extended clutching construction.

Extended clutching construction for the moduli of stable curves

Abstract

We give a description of the formal neighborhoods of the components of the boundary divisor in the Deligne-Mumford moduli stack of stable curves in terms of the extended clutching construction that we define. This construction can be viewed as a formal version of the analytic plumbing construction. The advantage of our formal construction is that we can control the effect of changing formal parameters at the marked points that are being glued. As an application, we prove that the infinitesimal neighborhood of the boundary component in is canonically isomorphic to the infinitesimal neighborhood of the zero section in the normal bundle. As another application, we show how to study the period map near the boundary components in terms of the coordinates coming from our extended clutching construction.

Paper Structure

This paper contains 17 sections, 22 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Changes of coordinates on a node deformation
  3. The gluing group scheme
  4. Isomorphisms between node deformations
  5. Extended clutching construction
  6. Construction
  7. Recovering the gluing data
  8. Computing the determinant line bundle
  9. The formal neighborhood of the boundary divisor and the gluing group scheme
  10. Changes of formal parameters on curves
  11. Canonical formal parameters in genus $1$
  12. Connection with a ${\mathbb G}_m$-torsor over ${\mathcal{E}}_{g_1,g_2}^{(n)}$
  13. Periods near a separating node
  14. Differentials and cohomology
  15. Basis of differentials regular outside a point
  16. ...and 2 more sections

Key Result

Theorem 1.1

The morphism $\xi_{g_1,g_2}^{(n)}$ (resp., $\xi_0^{(n)}$) factors through a morphism (resp., $[\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}/{\mathcal{G}}_{{\operatorname{glue}},n}]\to \Delta_0^{(n)}$). The restriction of this morphism, (resp., $[{\mathcal{M}}^{(\infty)}_{g-1,2}/{\mathcal{G}}_{{\operatorname{glue}},n}]\to U\Delta_0^{(n)}$) is canonically identified with the ${\mathbb G}_m$-torsor

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 34 more