Conic-connected Manifolds

Abstract

We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x^{\prime }\in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\le 2$. Moreover, a precise classification is obtained for $b_2=2$. Complete intersections of high dimension with respect to their multi-degree provide examples for the case $b_2=1$. The proof of the classification result uses a general characterization of rationality, in terms of suitable covering families of rational curves.

arXiv comment12 pages; v2: Theorem 1.3 with weaker formulation; more refined version of Theorem 1.5 (new numbering); more details proof Main Theorem 2.2;v3: Proposition 2.4 added. Final version to appear in Crelle's Journal