Conic-connected Manifolds

Details Conic-connected Manifolds Conic-connected Manifolds

2007-01-30

arxiv.org/abs/math/0701885v3

Mathematics

Algebraic Geometry

12 pages; v2: Theorem 1.3 with weaker formulation; more refined version of Theorem 1.5 (new numbering); more details proof Mai

Scientific paper

We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x' \in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\leq 2$. 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.

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