Rationality properties of manifolds containing quasi-lines

Abstract

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X' that contains a quasi-line, ie a smooth rational curve whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: There is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification X' which is strongly-rational We prove that strongly-rational varieties are stable under smooth, small deformations. Finally, we relate the previous results and formal geometry. This relies on \tilde{e}(X,Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion of X along Y. As applications we show various instances in which X is determined by this formal completion. We also formulate a basic question about the birational invariance of \tilde{e}(X,Y).