Unirationality of Fano Varieties

Fano Varieties of Low-degree Smooth Hypersurfaces and Unirationality

Unirationality and Existence of Infinitely Transitive Models

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of an algebraic group generated by unipotent algebraic subgroups. We expec...

On Birational Boundedness of Fano Fibrations

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of -Fano type. Furthermore, we show b...

Fano symmetric varieties with low rank

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are Fano. When G is semisimple we classify also the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the ...

Birational Automorphisms of Higher-dimensional Algebraic Varieties

The present survey covers the known results on the groups of birational automorphisms, rationality problem and birational classii-cation for Fano brations. 0. Birational geometry starts with M.NN other's paper 45] on Cremona transformations. The problems of birational geometry of algebraic varieties, that is, birational classiication, the rationality and unirationality problems, structure of th...