Algebraic Geometry Seminar

Title

The extension of numerically trivial divisors on a family.

Abstract

For a projective morphism f: X to S, we explore when it is possible to extend a divisor that is numerically trivial over an open subset to a global relatively num-trivial divisor. In particular, we show that such $L$ always exists after a (weak) semi-stable reduction when $\dim S=1$.

On the other hand, we give an example showing that $L$ may not exist (after any reasonable modification of $f$) if $\dim S\ge 2$, which also gives an $f_U$-nef divisor $M_U$ that cannot extend to an $f$-nef ($\mathbb{Q}$) divisor $M$ for any compactification of $f|_U$, even after replacing $X_U$ with any higher birational model.