Explicit constructions of Halphen pencils of index two
In the paper under review the author studies Halphen pencils and the associated rational elliptic surfaces. We say that a rational surface \(Y\) is a rational elliptic surface if \(Y\) admits a relatively minimal fibration \(f : Y \rightarrow \mathbb{P}^{1}\) whose generic fiber is a smooth curve of genus one. In this setting, there exists some \(m\geq 1\), called the index of the fibration so that \(f\) is given by the anti-pluricanonical system \(|-mK_{Y}|\). Moreover, \(m=1\) if and only if \(f\) admits a global section and whenever \(m>1\) there exists exactly one multiple fiber in \(Y\), this of multiplicity \(m\). Recall here that an Halphen pencil of index \(m\) is a pencil of plane curves of degree \(3m\) with nine (possibly infinitely near) base points of multiplicity \(m\). It is known that if \(f : Y \rightarrow \mathbb{P}^{1}\) is a rational elliptic surface with index \(m\) and \(F\) is a choice of a fiber of \(f\), then there exists a birational map \(\pi : Y \rightarrow \mathbb{P}^{2}\) such that \(f \circ \pi^{-1}\) is an Halphen pencil of index \(m\). Conversely, for a given Halphen pencil of index \(m\), taking the minimal resolution of its base points, we obtain a rational elliptic surface of index \(m\). The first main aim of the paper under review is to show that for each type of singular fibers that can occur one can construct at least one Halphen pencil yielding that type of fiber, and in some cases, one can construct all possible examples. This is done by analyzing plane curves corresponding to pencils. Recall here that any non-multiple fiber \(F\) corresponds to a plane curve \(B\) of degree \(3m\) that is not a multiple cubic. In the first step, the author studies singularities of a plane curve that occurs in a Halphen pencil via the log canonical threshold (lct for short). The main result in this direction establishes precise relations between the log canonical thresholds of the pairs \((Y,F)\) and \((\mathbb{P}^{2},B)\). Denote by \(M_{B}\), or resp. \(M_{F}\), the largest multiplicity of a component of \(B\), or resp. \(F\). Theorem A. If \(F\) is any non-multiple fiber of \(Y\), then the corresponding plane curve \(B\) is such that \[ \mathrm{lct}(\mathbb{P}^{2},B) \leq \frac{1}{M_{B}}\leq 2 \cdot \mathrm{lct}(Y,F), \] and these inequalities do not depend on the index \(m\) of the fibration. Furthermore, if \(m>1\) and \(F\) is reduced, then \(B\) is reduced and we have that \[ \frac{1}{m} < \mathrm{lct}(\mathbb{P}^{2}, B) \leq \mathrm{lct}(Y,F), \] and if \(M_{F} \geq m\) and \(F\) is not reduced, then \[ \mathrm{lct}(Y,F) \leq \mathrm{lct}(\mathbb{P}^{2},B). \] The second main result (a general one that does not depend on \(m\)) tells us the following. Theorem B. If \(F\) is of type \(II^{*}\), \(III^{*}\) or \(IV^{*}\) (according to Kodaira's classification), then \(B\) cannot be a reduced curve. Moreover, in the case when \(m=2\), the author provides a complete classification of plane curves \(B\) provided that \(F\) is of type \(II^{*}\), \(III^{*}\) or \(IV^{*}\).
scientific article; zbMATH DE number 7598573
scientific article