Number of singularities of stable maps on surfaces

Takahiro Yamamoto

Vol. 280 (2016), No. 2, 489–510

DOI: 10.2140/pjm.2016.280.489

Abstract

Let $N$ denote the plane ${ℝ}^2$ or the $2$-sphere $S^2$. In this paper, we determine the $5$-tuples of integers $\left(g,d,i,c,n\right)$ such that there exists a degree $d$ stable map ${\Sigma }_g\to N$ whose singular point set consists of $i$ connected components, $c$ cusps, and $n$ nodes, where ${\Sigma }_g$ is the standard genus $g$ surface.

Keywords

stable map, cusp, node

Mathematical Subject Classification 2010

Primary: 57R45

Secondary: 57M20, 58K15

Milestones

Received: 9 August 2014

Revised: 25 March 2015

Accepted: 3 June 2015

Published: 28 January 2016