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 DeligneMumford 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 1st infinitesimal neighborhood of the boundary component $\Delta_{1,g1}$ is canonically isomorphic to the 1st infinitesimal neighborhood in the normal bundle, near the locus ${\mathcal M}_{1,1}\times {\mathcal M}_{g1,1}\subset \Delta_{1,g1}$ corresponding to pairs of smooth curves with marked points. As another application, we show how to study the period map near the boundary components $\Delta_{g_1,g_2}$ in terms of the coordinates coming from our extended clutching construction.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.04682
 Bibcode:
 2021arXiv211004682P
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 23 pages