| Abstract |
Let $f\colon X\to\mathbb{P}^1$ be a smooth morphism of projective complex varieties. It was shown by Pieloch, using symplectic techniques, that $f$ always has a section. I will report on work in progress with Ben Church where we present two Hodge/MMP-theoretic criteria for a fibration onto a curve to have a section. In particular, these criteria give an algebraic proof of Pieloch's result assuming the Abundance Conjecture. Under a suitable ordinarity hypothesis we prove a positive characteristic version of the above results; furthermore, when this hypothesis fails, we show that there are smooth fibrations $X\to\mathbb{P}^1$ without sections. |
