On the classification of fibrations by genus two singular curves via fibrations by elliptic curves on surfaces

TL;DR

This paper classifies fibrations by genus two singular curves in positive characteristic, using smoothing techniques to relate them to elliptic fibrations on rational surfaces, revealing new insights into their structure.

Contribution

It introduces a classification method for genus two singular curve fibrations via smoothing to elliptic fibrations on rational surfaces, expanding understanding of fibrations in positive characteristic.

Findings

Classification of genus two singular curve fibrations using smoothing techniques

Identification of vector fields that recover fibrations via quotients of rational elliptic surfaces

Connection between singular fibrations and elliptic fibrations on rational surfaces

Abstract

In 1944 Zariski discovered that Bertini's theorem on variable singular points is no longer true when we pass from a field of characteristic zero to a field of positive characteristic. In other words, he found fibrations by singular curves, which only exist in positive characteristic. Such fibrations are connected with many interesting phenomena. For instance, the extension of Enrique's classification of surfaces to positive characteristic (Bombieri and Mumford in 1976), the counterexamples of Kodaira vanishing theorem (Mukai in 2013 and Zheng in 2016) and the isolated singularities with infinity Milnor number (Hefez, Rodrigues and Salom\~ao in 2019). In this work we are going to show that the smoothing process introduced by Shimada in 1991 can be used to classify the set of fibrations by genus two singular curves, up to isomorphism among their generic fibers, such that their smoothing are elliptic fibrations on rational surfaces. Moreover we will also describe the vector fields that can be used to recover such fibrations by singular curves via quotient of rational elliptic surfaces.