Number of singularities of stable maps on surfaces
type: Item
instance of: scholarly article
title: Number of singularities of stable maps on surfaces
zbMATH Open document ID: 1334.57031
author: Takahiro Yamamoto
published in: Pacific Journal of Mathematics
publication date: 2016-01-29T00:00:00Z
review text: Let \(\Sigma_{g}\) denote a surface of genus \(g\) and let \(N\) be either a 2-sphere \(S^2\) or the Euclidean Plane \( \mathbb{R}^2\). The author proves that for any stable map \(f\) : \(\Sigma_{g}\) \(\rightarrow\) \(N\) of degree \(d\) (wherever degree makes sense) having \(i\) components of the set of singular points, there are subsets \(L_{i,g}\), \(D\) and \(L_{i,g}\) \(\cap\) \(D\) of \( \mathbb{R}^2\) such that the cusps and nodes of the map \(f\) lie in the set \(L_{i,g}\) \(\cap\) \(D\). Conversely it is shown that if a pair of natural numbers \((c, n)\) \(\in\) \(L_{i,g}\) \(\cap\) \(D\), there is a stable map map \(f\): \(\Sigma_{g}\) \(\rightarrow\) \(N\) with singular set consisting of \(i\) components, \(c\) cusps and \(n\) nodes. Many other similar results regarding apparent contours of stable maps are obtained in this paper. The paper is very technical in nature, so some more detail in the description of the sets \(L_{i,g}\) and \(D\) could have made it easier to understand, however sufficiently many references are included for further information.
reviewed by: Keerti Vardhan
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 58K15
zbMATH DE Number: 6537062
zbMATH Keywords: stable map
zbMATH Keywords: cusp
zbMATH Keywords: node
zbMATH Keywords: germ map
MaRDI profile type: Publication
full work available at URL: pjm.2016.280.489
OpenAlex ID: W2289297174
DOI: 10.2140/PJM.2016.280.489
description: scientific article; zbMATH DE number 6537062