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arXiv · 0906.5388

Polarized minimal families of rational curves and higher Fano manifolds

Carolina Araujo·Ana-Maria Castravet
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Abstract

In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in X$. First we translate positivity properties of the Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows us to classify polarized minimal families of rational curves associated to Fano manifolds $X$ satisfying $ch_2(X)\geq0$ and $ch_3(X)\geq0$. As a first application, we provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to $\mathbb P^2$ and $\mathbb P^3$. Moreover, this classification enables us to find new examples of Fano manifolds satisfying $ch_2(X)\geq0$.

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BibTeXRIS

Carolina Araujo, Ana-Maria Castravet. 2009-06-30. Polarized minimal families of rational curves and higher Fano manifolds. https://arxiv.org/abs/0906.5388

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