Algebraic geometry and projective differential geometry, Seoul National University concentrated lecture series, 1997

ICM lecture on minimal models and moduli of varieties.

This text is a draft of the review paper on projectively dual varieties. Topics include dual varieties, Pyasetskii pairing, discriminant complexes, resultants and schemes of zeros, secant and tangential varieties, Ein theorems, applications of projective differential geometry and Mori theory to dual varieties, degree and multiplicities of discriminants, self-dual varieties, etc.

The aim of these notes is to give a introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves which are the higher dimensional analogs of rational curves and surfaces. The first six sections closely follow the lectures of J. Koll\'ar given at the summer school ``Higher dimensional varieties and rational points''. The notes were written up by C. Araujo and substantially edited later. Revised...

This is a detailed write-up of Koll\'ar's course at the EMS summer school in Algebraic Geometry in Eger, Hungary, August 1996. The topics include definitions and examples of rational and unirational varieties, with special attention to varieties defined over non-algebraically closed fields, Segre's theorem on non-rationality of cubic surfaces of Picard number one, Manin's theorem that birationally equivalent cubic surfaces of Picard number one are projectively equivalent, and...

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

This article is a transcription of a video of a 1972 lecture by Jean Dieudonn\'e, enhanced with composite still images from the video. The lecture covers the same material as an earlier paper and lecture notes by Dieudonn\'e, but the live lecture has a character of its own.

In the first part of the thesis, we study a classical invariant of projective varieties, the secant defectivity. The second part is devoted to modern algebraic geometry, we study the birational geometry of blow-ups of Grassmannians at points.

In this paper we prove some general results on secant defective varieties. Then we focus on the 4--dimensional case and we give the full classification of secant defective 4--folds. This paper has been inspired by classical work by G. Scorza,

I prove that every smooth legendrian variety generated by quadrics is a homogeneous variety and further I give a list of all such legendrian varieties. A review of the subject is included, illustrated by examples. Another result is that no complete intersection is a legendrian variety.

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012.