On the classification of fibrations by singular curves on unirational 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 (jointly work with Hefez and Rodrigues in 2019). In this talk we are going to show that the smoothing process introduced by Shimada in 1991 can be used to describe the set of fibrations by genus two singular curves on unirational surfaces, up to isomorphism among their generic fibers, such that the smoothing are elliptic fibrations. Moreover we will also describe the vector fields whose tangencies with elliptic fibrations generate such fibrations by singular curves, after the quotient of the rational elliptic surfaces. This is a work in progress with J. H. O. Rodrigues and R. O. C. Santos.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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