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 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, 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 caracterization of these varieties.Finally, we relate the previous results and formal geometry. This relies on e(X, Y ), a numerical invariant of a given quasi-line Y that depends only on the formal completion X| Y . As applications we show various instances in which X is determined by X| Y . We also formulate a basic question about the birational invariance of e(X, Y ).

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It is shown that, on a smooth surface, the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.

A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.The jumping numbers of a curve on a smooth complex surface are a sequence of positive rational numbers revealing information about the singularities of the curve. They extend in a natural way the information given by the log-canonical threshold, the smallest jumping number (see [3] for example). They are periodic, completely determined by the jumping numbers less than 1, but otherwise difficult to compute in general, even if a set of candidates is easy to provide, cf. [9, Lemma 9.3.16].The aim of this paper is to give a formula for the jumping numbers of a curve unibranch at a singular point. A curve C will be said to be unibranch at a point P , if the analytic germ of C at P is irreducible. The formula is expressed in terms of the Enriques diagram associated to the singularity, or equivalently (see Theorem 3.1) in terms of a minimal set of generators (β 0 , β 1 , . . . , β g ) of the semigroup S(C, P ) of C at P :where the m j are defined below. HereThe semigroup is defined by S(C) = {ord P s | s ∈ O C,P }, the order of the local section s being computed using a normalization of C. It is finitely generated and a minimal set of generators (β 0 , β 1 , . . . , β g ) is constructed as follows (see [13, Theorem 4.3.5]): β 0 is the least element of S(C); set m 1 = β 0 ; β j is the least element of S(C) not divisible by m j and m j+1 = gcd(m j , β j ).To prove (1) we use the notion of relevant divisors of the minimal log resolution of C at P , notion introduced in [12], and previously in [6] from the point of view of valuations corresponding to Puiseux exponents: a relevant divisor is an irreducible exceptional divisor that intersects at least three other components of the total transform of C through the resolution. When C is unibranch at P , we show that the relevant divisors account for all the jumping numbers. This is the content of Proposition 2.5 and represents the key step of the proof. In 1