An invitation to the local structures of moduli of genus one stable maps (Q2837136)

Let \(\overline M_g(\mathbb P^n,d)\) be the moduli space of genus \(g\) degree \(d\) stable maps to \(\mathbb P^n\). If \(g>0\), then this moduli space will in general be singular, of the wrong dimension, and the locus of smooth curves will fail to be dense. When \(g=1\) and \(d \geq 3\), the locus of smooth curves is nonsingular of the expected dimension. \textit{R. Vakil} and \textit{A. Zinger} [Geom. Topol. 12, No. 1, 1--95 (2008; Zbl 1134.14009)] constructed an explicit desingularization of its closure inside the space of stable maps by a sequence of blow-ups along boundary loci. \textit{Y. Hu} and \textit{J. Li} [Math. Ann. 348, No. 4, 929--963 (2010; Zbl 1201.14019)] gave a different, purely algebro-geometric treatment of Vakil-Zinger's construction.NEWLINENEWLINELet \(\pi : X \to \overline M_g(\mathbb P^n,d)\), \(f : X \to \mathbb P^n\) be the universal family. The sheaf \(\pi_\ast f^\ast \mathcal O(k)\) will in general fail to be locally free. However, if \((\pi',f')\) is the pullback of the universal family over Vakil-Zinger's moduli space, then \(\pi'{}_{\ast} f'{}^{\ast} \mathcal O(k)\) will in fact be locally free. In the terminology of the paper under review, they don't just resolve the moduli space, but also the sheaf.NEWLINENEWLINEThe current paper consists of a large number of explicit instructive examples describing the singularities that occur on the boundary of \(M_1(\mathbb P^n,d)\) and how they may be resolved. In this way it can serve as an introduction to and motivation for the papers of Ji--Hu and Vakil--Zinger. The viewpoint of ``resolving the sheaf'' is emphasized.NEWLINENEWLINEFor the entire collection see [Zbl 1245.00047].