Classification of quadruple Galois canonical covers II

Abstract

In this article we classify quadruple Galois canonical covers $\phi$ of singular surfaces of minimal degree. This complements the work done in math.AG/0302045, so the main output of both papers is the complete classification of quadruple Galois canonical covers of surfaces of minimal degree, both singular and smooth. Our results show that the covers $X$ studied in this article are all regular surfaces and form a bounded family in terms of geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4. Together with the results of Horikawa and Konno for double and triple covers, a striking numerology emerges that motivates some general questions on the existence of higher degree canonical covers. In this article, we also answer some of these questions. The arguments to prove our results include a delicate analysis of the discrepancies of divisors in connection with the ramification and inertia groups of $\phi$.