Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of script capital O signℙ1 (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, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion X̂Y. As applications we show various instances in which X is determined by X̂Y. We also formulate a basic question about the birational invariance of ẽ(X, Y).

Rationality properties of manifolds containing quasi-lines

IONESCU, Paltin;
2003

Abstract

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of script capital O signℙ1 (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, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion X̂Y. As applications we show various instances in which X is determined by X̂Y. We also formulate a basic question about the birational invariance of ẽ(X, Y).
2003
Ionescu, Paltin; D., Naie
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1682750
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