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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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