Families of canonically polarized manifolds over log Fano varieties

Let $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.

Manifolds Covered by Lines and Extremal Rays

Let X be a smooth complex projective variety, and let H ∈ Pic(X) be an ample line bundle. Assume that X is covered by rational curves with degree one with respect to H and with anticanonical degree greater than or equal to (dimX – 1)/2. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves NE(X).

A remark on elementary contractions

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.