This is a research project in the field of complex algebraic geometry around the following 2 problems: (i) proving there exists a Fano manifold that is not unirational, and (ii) finding conditions on the geometric generic fiber of a surjective morphism to a surface guaranteeing that the only obstruction to existence of a rational section is the Brauer obstruction. Although these projects seem quite different, they both involve the study of spaces of rational curves on varieties in a crucial way. In the first problem, one can prove that a Fano manifold is not unirational by proving there are "few" rational curves on the space of rational curves on the manifold. In the second problem, conjecturally, the condition is that the spaces of rational curves on the geometric generic fiber are themselves rationally connected.
In complex algebraic geometry, the notion of a variety defined over a non-algebraically closed field is the common situation of a sequence of polynomial equations in variables x,y,z,... that depend on parameters s,t,u,... (which themselves may satisfy some polynomial equations). A basic question is whether for each choice of parameters there is a solution of the equation of the form: x,y,z,... are quotients of polynomials in the parameters s,t,u,... This is the problem of rational points. Another question, in the case where the polynomials don't depend on parameters, is whether x,y,z,... can be written as polynomials in a set of free variables, a,b,c,... such that every choice of a,b,c,... gives a solution of the polynomials, and essentially every solution of the polynomials arises in this way. This is the problem of unirationality. Both are old, difficult problems with applications in algebraic geometry and number theory. This project applies new results about spaces of rational curves to each of these projects.
