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Almost-lines and quasi-lines on projective manifolds. (English) Zbl 1078.14010
Peternell, Thomas (ed.) et al., Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter (ISBN 3-11-016204-0/hbk). 1-27 (2000).
Let $$X$$ be a complex projective manifold of dimension $$n\geq 2$$, and let $$Y$$ be a smooth closed (connected) subvariety of $$X$$. For every integer $$m\geq 0$$, the $$m$$th infinitesimal neighborhood of $$Y$$ in $$X$$ is denoted by $$X(m)$$. According to the scheme structure of $$X(m)$$ there is a natural restriction homomorphism $$\alpha_m: \text{Pic}(X)\to \text{Pic}(X(m))$$ of Picard groups, whose properties should be of just as natural great interest.
As $$X(0)= Y$$, the first non-classical case would be the case of $$m= 1$$, and the question of under which conditions the restriction map $$\alpha_1$$ is surjective has indeed been studied, to quite some extent, by several authors in the past. The particular case of a curve $$Y$$ with positive self-intersection in a smooth projective surface $$X$$ has been completely settled by J. d’Almeida [Enseign. Math., II. Sér. 41, No. 1–2, 135–139 (1995; Zbl 0862.14023)], who showed that the map $$\alpha_1$$ is surjective essentially when $$Y$$ is a line in $$\mathbb{P}^2$$. In the paper under review, the authors generalize d’Almeida’s theorem to the case of a smooth connected curve $$Y$$ in a projective manifold $$X$$ of arbitrary dimension $$n\geq 2$$. Their main result states that the map $$\alpha_1$$ is surjective if and only if the following three conditions are satisfied:
(1) $$Y\cong\mathbb{P}^1$$, (2) $$N_{Y|X}\cong(n- 1){\mathcal O}_{\mathbb{P}^2}(1)$$, and (3) the Lefschetz map $$\alpha_0: \text{Pic}(X)\to \text{Pic}(Y)$$ is surjective.
In view of this theorem, the authors call a curve $$Y\subset X$$ a quasi-line if it satisfies conditions (1) and (2). If all three conditions (1), (2) and (3) hold for $$Y$$ then the curve $$Y$$ is called an almost-line. It is shown that in dimension $$n\geq 3$$ a quasi-line or almost-line need not be a line in $$\mathbb{P}^n= X$$, and therefore a thorough study of the geometry of quasi-lines and almost-lines is carried out in the second part of the article. The authors prove that projective manifolds containing quasi-lines and/or almost-lines are very special examples of rationally connected manifolds in the sense of Campana-Kollár-Miyaoka-Mori, and that these manifolds are stable under small deformations. Finally, in this context, a very special class of rational manifolds, which the authors call strongly rational, is introduced and analyzed in terms of properties of quasi-lines.
For the entire collection see [Zbl 0933.00031].

##### MSC:
 14C22 Picard groups 14J99 Surfaces and higher-dimensional varieties 14B10 Infinitesimal methods in algebraic geometry 14M20 Rational and unirational varieties