Let X be a smooth complex Fano variety. We study ‘quasi-elementary’ contractions of fiber type of X, which are a natural generalization of elementary contractions of fiber type. If f:X→Y is such a contraction, then the Picard numbers satisfy ρX≤ρY+ρF, where F is a general fiber of f. We show that, if dim Y ≤3 and ρY≥4, then Y is smooth and Fano; if moreover ρY≥6, then X is a product. This yields sharp bounds on ρX when dim X=4 and X has a quasi-elementary contraction of fiber type, and other applications in higher dimensions.

Lattès and Kummer examples are rational transformations of compact kähler manifolds that are covered by an affine transformation of a compact torus. We present a few ergodic characteristic properties of these examples. The main results concern the case of surfaces.

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

The Nagata automorphism is a kind of complicated automorphism on the affine 3-space . For a long time, it remained unknown whether or not the Nagata automorphism is tame until Shestakov and Umirbaev at last proved that it is not tame in 2004, by purely algebraic methods (e.g. Poisson algebra). In this paper, we consider a certain necessary condition for a given automorphism on to be tame from the point of view of the Sarkisov program established by Corti. Furthermore, by using it, we shall give a new algebro-geometric proof of the non-tameness of the Nagata automorphism.

For each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.