| Περίληψη: |
We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function is upper semicontinuous in flat families with reduced and irreducible fibers. Given |$\epsilon>0$| , we also prove that for a |$\mathbb{Q}$| -Cartier divisor |$B$| on a family of varieties |$X\rightarrow T$| , if |$(X_{t},|B_{t}|_{\mathbb{Q}})$| is |$\epsilon $| -lc and the volume of |$B_{t}$| is constant for densely many closed points |$t\in T$| , then the generic fiber |$(X_\eta ,|B_\eta |_{\mathbb{Q}})$| is also |$\epsilon $| -lc. Throughout this paper, the ground field is |$\mathbb{C}$|. By a variety , we mean an integral, separated scheme of finite type over a field (possibly non-algebraically closed). [ABSTRACT FROM AUTHOR] |