On local stable reduction of singularity \((y^a-x^b)(y^p-x^q)\) (Q1771529)

The authors consider a one-dimensional family of curves with smooth fibers except for the central fiber \(C_0\), which has a single singularity that is topologically equivalent to \((y^a-x^b)(y^p-x^q)\). They perform explicit blowups and base changes to obtain the stable reduction of such a family. The central fiber of the resulting family of stable curves consists of three components: the normalization \(C\) of \(C_0\) and two other curves \(\overline{E}\) and \(\overline{F}\). It is shown that \(C\) meets \(\overline{E}\) in \((p,q)\) points, that \(C\) meets \(\overline{F}\) in \((a,b)\) points, and that \(\overline{E}\) and \(\overline{F}\) meet in \((q,a)\) points. Explicit formulas are given for the genera of \(\overline{E}\) and \(\overline{F}\).