On the topological index of a singular fiber in a family of curves
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In this paper, we classify the singular fibers $F$ according to their topological index ${\rm Sign}(F)$ . As an application, we provide a sharp upper bound on the topological signature ${\rm Sign}(S)$ of a complex algebraic surface $S$ with a genus $g\geq 2$ fibration $f: S\rightarrow \mathbb{P}^1$, where $f$ has two singular fibers. We show that if ${\rm Sign}(S) \neq 0$, then ${\rm Sign}(S) \leq -4$.
Abstract
In this paper, we classify the singular fibers $F$ according to their topological index ${\rm Sign}(F)$ . As an application, we provide a sharp upper bound on the topological signature ${\rm Sign}(S)$ of a complex algebraic surface $S$ with a genus $g\geq 2$ fibration $f: S\rightarrow \mathbb{P}^1$, where $f$ has two singular fibers. We show that if ${\rm Sign}(S) \neq 0$, then ${\rm Sign}(S) \leq -4$.