# Rationality of Algebraic Varieties

# Rationality of Algebraic Varieties

Finding a rational parametrization for a system of polynomial equations has been studied for a long time, and it leads to the concept of rational varieties. Bézout’s theorem implies that degree 1 and 2 hypersurfaces are rational. It is a hard problem to determine whether a general variety is rational. For a smooth projective variety to be rational, it is necessary that any (tensor power of) differential form must vanish. This condition is also sufficient for curves and surfaces due to Riemann and Castelnuovo respectively. However, this is not true for higher dimensional varieties by the works of Clemens-Griffiths, Iskovskikh-Manin, Artin-Mumford, Kollár, etc.. In this talk, I will give an introduction to the rationality problem and prove the unirationality of cubic threefolds. If time permits, I will also explain some connections to differential geometry.