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Conic-connected manifolds. (English) Zbl 1200.14078
Let $$X$$ be a complex projective manifold that is rationally connected, i.e., two general points can be joined by a rational curve. Rationally connected manifolds are natural generalisations of unirational and Fano manifolds; studying their geometry in terms of a family of rational curves joining two general points is a natural, but quite difficult problem.
In the paper under review, the authors consider the case of an embedded manifold $$X \subset \mathbb P^n$$ such that two general points can be joined by a smooth conic. They prove that in this case $$X$$ is a Fano manifold with second Betti number $$b_2$$ at most two. If $$b_2=2$$, the manifold $$X$$ belongs to a short list of well-known examples: Segre embedded products of projective spaces and their hyperplane sections, or projections from a linear space of the Veronese variety $$v_2(\mathbb P^n)$$. If $$b_2=1$$ the authors do not obtain a complete classification but they show that $$X$$ is a Veronese variety or has index at least $$(\dim X+1)/2$$. This bound on the index is sharp, as one can easily construct complete intersections of index at least $$(\dim X+1)/2$$ that are conic-connected [cf. L. Bonavero and A. Höring, Acta Math. Vietnam. 35, No. 1, 23–30 (2010; Zbl 1204.14025)]. A key ingredient in the proof is an “infinitesimal rationality criterion”: a projective manifold is rational if and only if it admits a covering family of rational 1-cycles, all passing through $$x$$ and smooth at $$x$$ such that the general cycle is determined by its tangent vector at $$x$$. Further applications of this theorem can be found in the authors’ recent work on varieties with quadratic entry locus, P. Ionescu and F. Russo [Compos. Math. 144, No. 4, 949–962 (2008; Zbl 1149.14041)].

##### MSC:
 14J40 $$n$$-folds ($$n>4$$) 14E30 Minimal model program (Mori theory, extremal rays) 14J45 Fano varieties 14M10 Complete intersections
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##### References:
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