From the introduction: Let \(\pi:X\to D\) be a proper surjective holomorphic map of a complex manifold \(X\) of dimension 2 to a small open disk \(D= \{t\in\mathbb{C} \mid|t|< \varepsilon\}\). We assume that \(\pi\) is smooth over a punctured disc \(D'=D-\{0\}\). Moreover we assume that for every \(t\in D'\) the fiber \(X_t=\pi^{-1} (t)\) is a nonsingular curve of genus \(g\) and that \(X\) contains no exceptional curves of the first kind. By \(L_t\), we denote the effective divisor in \(X\) defined by the equation \(\pi=t\) \((t\in D)\). We call the divisor \(L_0\) the singular fiber of \(\pi\). For every \(t\in D'\) we call the divisor \(L_t\) a generic fiber.
In the study of elliptic surfaces, Kodaira showed that there exist only ten types of singular fibers of pencils of curves of genus one. Iitaka and Ogg gave a numerical classification of singular fibers of curves of genus 2. Namikawa and Ueno classified their numerical types completely, constructed all their singular fibers and calculated the monodromies around them. In a previous paper [K. Uematsu, Jap. J. Math., New Ser. 23, No. 2, 281-301 (1997; Zbl 0905.14006)], the author studied the numerical properties of singular fibers in pencils of curves of genus \(g\) \((\geq 2)\) and gave a method to classify all the numerical types of the singular fibers. In this article, by using this method, we give the complete numerical classification of singular fibers in pencils of curves of genus three. If the number of irreducible components is more then one, we have \(\Gamma^2 <0\) and \(\Gamma\cdot K_X\geq 0\), where \(K_X\) is a canonical divisor. If \(\Gamma\cdot K_X>0\), we call this component \(\Gamma\) a trunk. If \(\Gamma\cdot K_X=0\), then we have \(\Gamma^2 =-2\). Thus we call this component \(\Gamma\) a \((-2)\)-curve. Further we call a connected component consisting of \((-2)\)-curves in a singular fiber a branch. The method of numerical classification of singular fibers in the paper under review is as follows.
First, the author determines all the combinations of trunks, the number of which is finite. Secondly, he classifies the possible branches in the singular fiber in pencils of curves of genus 3. Finally by combining trunks with branches, he classifies all the singular fibers. In his calculation there exist 4343 types of singular fibers in pencils of curves of genus 3, while on the other hand, in the case of genus 2 there exist 140 types.