Rationality and Irrationality in Families of Varieties

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When can we write down all the solutions of a polynomial equation? We seek equations that can be parametrized with rational functions. These are used in mapmaking (stereographic projection), computer graphics, and modeling problems. Indeed, parametrizations are often the most efficient way to render geometric objects as screen images. Mathematicians have developed a rich theory for deciding when such parametrizations are possible. This project will advance this theory.An algebraic variety is rational if it can be obtained from projective space by a sequence of algebraic modifications. Given a family of smooth complex projective varieties, it is difficult to say which members are rational or irrational, or even to formulate qualitative results about the locus of rational members. The PI will use the technique of decomposition of the diagonal, and related tools from deformation theory, Hodge theory and classical geometry, to shed light on this question. The PI and his collaborators have recently shown that rationality is not a deformation invariant property: There are families of smooth complex projective varieties with both rational and irrational members. Despite examples, like cubic fourfolds, that have been extensively studied, there are no cases where the loci of rational and irrational members have been precisely described. The PI will refine and make effective the decomposition of the diagonal technique to clarify which members of a family are irrational - can they be expressed in Hodge-theoretic terms? At the same time, the PI will advance constructive techniques to exhibit rational and unirational parametrizations with prescribed properties.