Fibrations by curves with more than one nonsmooth point

Salomão, Rodrigo

doi:10.1007/s00574-014-0048-8

Cited by 3 publications

(1 citation statement)

References 17 publications

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“…Several papers studied the classification of fibrations by singular curves from the birational perspective as [29], [30], [23], [24] and [4]. All of them used the well known strategy -that in this context appeared at first in the work of Stöhr (see [29]) andwhich we outline below.…”

Section: Introductionmentioning

confidence: 99%

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

Rodrigues¹,

Salomão²,

Santos³

2023

Preprint

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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 Salomão 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.

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