Smooth minimal genera for small negative classes in \({\mathbb C}P^{2}{\#}n{\overline{{\mathbb C}P}^2}\) with \(n {\leqslant} 9\)
This is a contribution to the minimal genus problem for homology classes \(\xi\in H_2({\mathbb CP}^2 {\#} n\overline{\mathbb CP^2}; \mathbb Z)\), where \(\mathbb CP^2\) is the \(2\)-dimensional complex projective space. The problem is to find the minimal genus of a smoothly embedded orientable surface representing the class \(\xi\). For \(\xi^2\geq 0\), some results (including the complete solutions for some values of \(n\)) were previously achieved, e.g., by the same authors in [Topology 37, 575-594 (1998; Zbl 0899.57022)]. In the paper under review, the authors solve the minimal genus problem completely for \(\xi\in H_2({\mathbb CP}^2 {\#} n\overline{\mathbb CP^2}; \mathbb Z)\) such that \(0>\xi^2>-16\), when \(n\leq 9\). They also show that in \(H_2({\mathbb CP}^2 {\#} n\overline{\mathbb CP^2}; \mathbb Z)\) with \(1\leq n\leq 9\), all classes \(\xi\) with \(0>\xi^2>-n-7\) have minimal genus zero.
scientific article; zbMATH DE number 1961612
scientific article