Let $X$ be projective variety. Consider $\pi:X\to B$ be a projective morphism and take $$\mathcal D=\{b\in B|K_{X_b} \; \; \text{is semi-ample}\}$$ Then when $D$ is open and closed ?
1
-
5$\begingroup$ Since $K_{X_b}$ is semi-ample, hence it is nef. Let $A$ be an $\pi$-ample line bundle on $X$ and let $\sigma(X_s,K_{X_b},A)=min(AC|C\; \text{effective 1-cycle with}\; K_{X_b}C<0)$. The openness of $\mathcal D$ is equivalent to $\{(X_b,K_{A_b},A_b)\}_{b\in B}$ being bounded for some $A$. See Theorem 1 in gdz.sub.uni-goettingen.de/pdfcache/PPN365956996_0075/…. I don't have any idea about closedness $\endgroup$– user21574Commented Jul 30, 2017 at 2:56
Add a comment
|