A birational contraction of genus 2 tails in the moduli space of genus 4 curves I
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.14033
author: Donghoon Hyeon
author: Yongnam Lee
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.3973
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 6340385
zbMATH Keywords: moduli of curves
zbMATH Keywords: log canonical model
zbMATH Keywords: Hassett-Keel program
MaRDI profile type: Publication
OpenAlex ID: W2329210756
arXiv ID: 1003.3973
DOI: 10.1093/IMRN/RNT042
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published in: IMRN. International Mathematics Research Notices
title: A birational contraction of genus 2 tails in the moduli space of genus 4 curves I
review text: Brendan Hassett initiated the fascinating program of applying the minimal model program to the moduli space of stable curves by computing the sequence of log canonical models of \(\overline{\mathcal{M}}_g\) associated to a rational multiple \(\alpha\) of the total boundary divisor and studying the flips and divisorial contractions arising in this process. The spaces NEWLINE\[NEWLINE\overline{\mathcal{M}}_g(\alpha) := \mathrm{Proj} \bigoplus_{m \geq 0} H^0(\overline{\mathcal{M}}_g,m(K+\alpha D))NEWLINE\]NEWLINE start when \(\alpha=1\) by recovering the Deligne-Mumford compactification, since a classic result is that \(D\) is ample on \(\overline{\mathcal{M}}_g\), then one decreases \(\alpha\) as far as one can. For large enough genus (e.g., \(g = 22\) or \(g \geq 24\)) this moduli space is of general type so the sequence terminates with \(\alpha = 0\) being the canonical birational model of \(\overline{\mathcal{M}}_g\), a tantalizing object believed to be a moduli space of highly singular curves but whose basic properties and concrete description remain a mystery. This program is often called the Hassett-Keel program. For small genus the log canonical models end up being a point long before \(\alpha\) reaches zero, so rather than describing the (uninteresting) canonical model, the goal is simply to describe the entirely of the finitely many steps in the program until the last model is indeed a point. In fact, Hassett launched this program with his wonderful paper [\textit{B. Hassett}, Prog. Math. 235, 169--192 (2005; Zbl 1094.14017)] detailing the case \(g=2\). The big results then came in papers of \textit{B. Hassett} and \textit{D. Hyeon} [Trans. Am. Math. Soc. 361, No. 8, 4471--4489 (2009; Zbl 1172.14018), and Ann. Math. (2) 177, No. 3, 911--968 (2013; Zbl 1273.14034)] which described the first divisorial contraction and the first flip in all genera.NEWLINENEWLINESubsequently, work on the program bifurcated in some sense. Some authors wanted to push the complete d…
reviewed by: Noah Giansiracusa
description: scientific article; zbMATH DE number 6340385