Università Roma Tre

The Course aims to introduce, in an elementary way and building on algebraic and geometric tools offered to the student in previous courses, those algebraic varieties which admit rational parametric equations. A sequel of relevant examples, both from the geometrical and historical points of view, will be useful to describe the contemporary state of the art on rational parametric equations and, more specifically, on what is known, or not known, on the rationality problem for some typical classes of algebraic varieties.
The Course will consider the following topics:
(1) Curves and rationality: Lueroth problem.
(2) Embeddings of the projective line.
(3) Projective varieties of minimal degree.
(4) Del Pezzo surfaces.
(5) Rationality criterion for algebraic surfaces.
(6) (Uni)rationality problems for hypersurfaces.
(7) Lueroth problem.
(8) Cubics.
(9) Artin-Mumford's counterexample.
(10) Outline of Vosin's method.

Text;
A. Corti, J. Kollar, K. Smith 'Rational and nearly rational varieties', Cambridge studies in advanced mathematics vol. 92, Cambridge UP (2004)

Some reference books for the course:
1) A. Beauville, B. Hassett, A. Kuznetsov, A. Verra 'Rationality problems in Algebraic Geometry' Lecture Notes in Mathematics vol. 2172, Springer (2016)
2) I. Dolgachev 'Classical Algebraic Geometry. A Modern View' Cambridge UP (2016)
3) M. Mustata 'Topics in Algebraic Geometry II. Rationality of Algebraic Varieties', Note on line: https://web.math.princeton.edu/~takumim/Rationality.pdf