Number of singularities of stable maps on surfaces
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.
scientific article; zbMATH DE number 6537062