On confluence of singular fibers in elliptic fibrations
In a deformation of elliptic surfaces, one often observes several singular fibres \(\{F_ i\}\) with \(1\leq i\leq r\) of nearby surfaces ``flowing'' (see \(definition\quad 2.2)\) together into one singular fibre F. Such a phenomenon is called a confluence of singular fibres. In the special case where \(r=3\), let \(F_ 1\), \(F_ 2\) and \(F_ 3\) be of type \(I_ a\), \(I_ b\) and \(I_ c\) in the sense of Kodaira and let \(M_ i\) with \(1\leq i\leq 3\) \((resp.\quad M)\) be the monodromy matrices associated to \(F_ i\) \((resp.\quad F),\) the author shows that this confluence phenomenon leads to a Diophantine equation of the form \(bcx^ 2+acy^ 2+abz^ 2-abcxyz=d\) \((d:=2-tr(M)\) where x, y and z are the SL(2,\({\mathbb{Z}})\) invariants associated to \(M_ 1\), \(M_ 2\) and \(M_ 3\). However, these invariants depend on the choice of the path taken to define the matrices \(M_ i\), so in this paper the author tries to classify the triple \((M_ 1,M_ 2,M_ 3).\) A generalization to the cases where \(r>3\) is discussed.
scientific article; zbMATH DE number 4083810