Rationality of Varieties and Related Topics

Dear Colleagues,

An algebraic variety is rational if it is birational to a projective space of some dimension. Thus, rational varieties can be thought of as the simplest algebraic varieties. The question whether a variety is rational and, in the affirmative case, the construction of one of its rational parametrizations goes back to the beginnings of algebraic geometry and has led to many challenging problems. A weaker notion of rationality is that of unirationality. A variety is unirational if it is rationally dominated by a projective space, that is, if it can be parametrized by rational functions. The problem whether unirationality implies rationality is a classical problem, known as the Lüroth problem. It has an affirmative answer for curves (J. Lüroth, 1876), as well as for complex surfaces (G. Castelnuovo, 1893). However, this is no longer true in higher dimensions. One of the first counterexamples is due to C. Clemens and P. Griffiths (1972), who showed that a smooth complex cubic hypersurface, which is known to be unirational, is not rational. The notion of unirationality also plays a crucial role in the theory of moduli spaces of varieties. In fact, from a unirational parametrization of the moduli space M (especially when this can be described algorithmically) we can construct the generic member of M.

Dr. Giovanni Staglianò
Guest Editor

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• rational variety
• birational map
• rational parametrization
• unirational variety
• unirationality for moduli spaces