2022/05/30 by Dubouloz, Adrien
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.15098
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves B in smooth projective surfaces X defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces X ∖ B admit infinitely many families of \mathbbA1-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For \mathbbA1-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence classes of such fibrations up to composition by automorphisms at the source and target is finite if and only if the self-intersection number of B in X is less than or equal to 6.