Birational Geometry and Rational Connectedness

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This project concerns birational geometry of complex varieties and the study of rationally connected varieties. The main objective is to study spaces of rational curves on rationally connected varieties, with the goal of constructing new birational invariants of these varieties and determining the relationship between these varieties and unirational varieties. The secondary objective is to develop further some of the techniques and tools of algebraic geometry: to develop techniques for determining when moduli stacks are rationally connected, to construct and study new compactifications of the space of smooth rational curves in projective space, and to refine the Behrend-Manin stratification of the Kontsevich moduli stack so that it detects mutliple covers.Systems of polynomial equations in some collection of variables arise in every branch of mathematics, science and engineering. Determining the solutions of these systems of equations is of paramount importance. "Unirational varieties" precisely correspond to those systems of equations where the set of solutions is most easily described -- the set of solutions is the set of all outputs of some sequence of polynomials with arbitrary inputs. So it is very important to recognize which systems of equations correspond to unirational varieties. From the point of view of geometry, a closely related notion is "rational connectedness". This notion is far simpler to recognize in practice. Potentially both notionsare equivalent. If one could prove that both notions areequivalent, this would give a simpler way to recognize which systems of equations correspond to unirational varieties. The investigator intends to continue his research of this problem.