Special birational transformations of projective spaces

Birational geometry of hypersurfaces in products of projective spaces

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P × P, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using this, we give new simple examples of non-Mori dream spaces and analogues of Mumford’s example of a strictly nef line bundle which is not ample.

HyperKähler Manifolds and Birational Transformations

The minimal model program has greatly enhanced our knowledge on birational geometry of varieties of dimension 3 and higher. About the same time, the last two decades have also witnessed increasing interests in HyperKähler manifolds, a particular class of Calabi-Yau manifolds. One interest in this area, which we hope to treat in the future, is to investigate the behavior of the SYZ mirror conjec...

On Certain Groups of Birational Contact Transformations*

On birational monomial transformations of plane

We study birational monomial transformations of the form φ(x : y : z) = (ε1x1y1z1 : ε2x2y2z2 : xα3yβ3zγ3), where ε1,ε2 ∈ {−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the transformationφ as a product of generators of the group. To prove this formula, we use birati...

Special linear groups generated by transvections and embedded projective spaces

We give a characterization of the ‘special linear groups’ T (Ψ,W ) as linear groups generated by a non-degenerate class Σ of abstract root groups such that the elements of A ∈ Σ are transvections.