2014/08/06 by Dubouloz, Adrien, Kishimoto, Takashi
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1408.1328
We show that the generic fiber of a family of smooth \mathbbA1-ruled affine surfaces always carries an \mathbbA1-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an \mathbbA1-fibration over a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational \mathbbA1-ruled surfaces, this induced \mathbbA1-fibration can also be obtained from a relative Minimal Model Program applied to a relative smooth projective model of the family.