2020/12/01 by Wang, Long
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.00272
We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well. The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi--Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:= ℙn × ⋯ × ℙn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1, …, 1) in P with dim X ≥ 3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.