The extension of numerically trivial divisors on a family
Lingyao Xie
TL;DR
The paper investigates when a relatively numerically trivial divisor on a family extends to a global divisor on the total space. It shows that for a family over a curve that becomes (weakly) semistable, one can, after a finite base change, extend up to a positive multiple to a globally defined divisor with relative numerical triviality, using Picard schemes and Néron models. However, in higher-dimensional bases such extensions fail in general, as illustrated by explicit counterexamples arising from the universal genus one family with two marked points, where even nef extensions are obstructed. The results illuminate the role of Pic^0 degeneration and Néron models in extending line bundles across families and delineate the limits of such extensions in higher dimensions.
Abstract
Let $f:X\to S$ be a projective morphism of normal varieties. Assume $U$ is an open subset of $S$ and $L_U$ is a $\mathbb{Q}$-divisor on $X_U:=X\times_S U$ such that $L_U\equiv_U 0$. We explore when it is possible to extend $L_U$ to a global $\mathbb{Q}$-divisor $L$ on $X$ such that $L\equiv_f 0$. 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.