Polarized minimal families of rational curves and higher Fano manifolds (Q2879625)

A higher Fano manifold is a Fano manifold \(X\) whose Chern characters satisfy some positivity conditions.NEWLINENEWLINEThis notion has been introduced to approach the problem of finding rational sections for rationally connected fibrations over surfaces or higher dimensional varieties; in this context various definitions of rationally simply connectedness have been introduced. Roughly speaking a smooth projective variety is RSC if a suitable irreducible component of the space of rational curves through two points of \(X\) is rationally connected.NEWLINENEWLINEIn the paper under review the authors consider Fano manifolds such that \(\text{ch}_2(X)>0\), i.e. \(\text{ch}_2(X) \cdot S >0\) for every surface \(S \subset X\). With the aim of studying the relation of this assumption with a reasonable notion of RSC, the authors undertake the task of understanding the properties of minimal familes of rational curves on such Fano manifolds.NEWLINENEWLINEGiven a Fano manifold \(X\), a general point \(x \in X\) and denoted by \(H_x\) a smooth and proper irreducible component of the scheme parametrizing rational curves passing through \(x\), with the polarization \(L_x\) given by the pullback of \(\mathcal O(1)\) via the finite ``tangent map'' \(\tau_x: H_x \to \mathbb P(T_xX^\vee)\), positivity conditions of \(\text{ch}_k(X)\) are translated into positivity conditions for \(\text{ch}_{k-1}(H_x)\) and into properties of the polarized pair \((H_x, L_x)\).NEWLINENEWLINEA remarkable result obtained with these techniques shows that, if \(X\) is a Fano manifold with \(\text{ch}_2(X)>0\) then \(H_x\) is a Fano manifold with large index (for the precise statement see Theorem 1.4). A second result (Theorem 1.5) relates positivity properties of \(\text{ch}_2(X)\) and \(\text{ch}_3(X)\) with the existence of a rational surface or a rational threefold through a general point of \(X\).NEWLINENEWLINEIn the final section, the authors discuss ideas giving evidence that the positivity of \(\text{ch}_2(X)\) is related to the existence of very twisting scrolls, a common requirement of the various definitions of RSC.NEWLINENEWLINEIn addition to the above mentioned results, the paper contains a noteworthy sections dedicated to examples of Fano manifolds with \(\text{ch}_2(X)\geq 0\).