We study singularities of algebraic varieties, in particular those arising in birational
geometry, from several points of view. The first is that of arc schemes: arc schemes …

These lecture notes have been prepared for the Summer school on” Moduli spaces and arcs
in algebraic geometry”, Cologne, August 2007. The goal is to explain the relevance of …

The multiplicity of a point on a variety is a fundamental invariant to estimate how the
singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can …

Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a
closed subscheme of X. We are interested in invariants attached to the singularities of the …

We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety
$ X $(over a field of characteristic zero). For this we introduce the notion of\emph {formal …

Boundedness and moduli of algebraic varieties Page 1 Surveys in Differential Geometry XXVI
Boundedness and moduli of algebraic varieties Caucher Birkar Abstract. In this short note we …

The study of singularities of pairs is fundamental for higher dimensional birational geometry.
The usual approach to invariants of such singularities is via divisorial valuations, as in [Kol] …

This thesis concerns the Birational geometry of Fano varieties. The first two chapters are an
introduction to Birational geometry, and then specifically the theory of Birational rigidity …

The aim of this work is to simplify the definitions related to the study of singularities of normal
varieties initiated in and. We introduce a notion of discrepancy for normal varieties, and we …

This is a preliminary draft of monograph. It builds on lectures notes on a course that
Lawrence Ein gave at the University of Catania in Summer 1998, and later again at Hong …