Contractions of subcurves of families of log curves

Abstract

Let C be a nodal curve, and let E be a union of semistable subcurves of C. We consider the problem of contracting the connected components of E to singularities in a way that preserves the genus of C and makes sense in families. In order to do this, we introduce the notion of mesa curve, a nodal curve C with a logarithmic structure and a piecewise linear function $\overline{\lambda }$ on the tropicalization of C. We then construct a contraction of the support of $\overline{\lambda }$ inside of C for families of mesa curves.

Keywords:

• Algebraic geometry; moduli of curves; logarithmic geometry; tropical geometry

2020 Mathematics Subject Classification:

• 14H10; 14H2; 14T05

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Acknowledgements

The content of this article forms the first part of my doctoral thesis at the University of Colorado at Boulder. I would like to thank my advisor, Jonathan Wise, for his inexhaustible patience and guidance, without which this work would not be possible. I would also like to acknowledge Dan Abramovich, Luca Battistella, Renzo Cavalieri, Francesca Carocci, Qile Chen, Andy Fry, Brendan Hassett, Leo Herr, Keli Parker, Dhruv Ranganathan, Hanson Smith, David Smyth, and John Willis for their interest, encouragement, and many helpful conversations.