Apparent contours of stable maps between closed surfaces

Abstract

Let $M$ and $N$ be connected and orientable, closed surfaces. For a stable map $\varphi : M \rightarrow N$, denote by $c(\varphi)$ and $n(\varphi)$ the numbers of cusps and nodes of $\varphi$ respectively. In this paper, we determine the minimal number $c(\varphi) + n(\varphi)$ among the apparent contours of degree $d$ stable maps $M \rightarrow N$ whose singular points set consists of one component.