Birational transformations simplifying singularities of algebraic curves

The Reduction of Singularities of Plane Curves by Birational Transformation

Singularities of Integrable Systems and Algebraic Curves

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the correspo...

Birational invariants of algebraic varieties

We extend the notion of Iitaka dimension of a line bundle, and deene a family of dimensions indexed by weights, for vector bundles of arbitrary ranks on complex varieties. In the case of the cotangent bundle, we get a family of birational invariants, including the Kodaira dimension and the cotangent genus introduced by Sakai. We treat in detail several classes of examples, such as complete inte...

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...

Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves

We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is give...