3 problems

Let $B$ be a generalized del Pezzo surface with $b_2(X)=b_2(B)+1$, and let … be a smooth genus-one fibered Calabi–Yau threefold with a 5-section. Assume that $X$ is the degeneracy…

Kollár's conjecture. $X$ is genus one (or elliptically) fibered if and only if there exists a divisor $D\in H^{2}(X,\mathbb{Q})$ satisfying

Let a Calabi–Yau variety admit a relatively minimal genus-one fibration with multiple fibers, and let a multiple fiber have multiplicity $m$. Discrete-torsion conjecture. The multi…