Graded rings and polarised varieties

Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties.