Fano manifolds with many free divisors
Let \(X\) be an \(n\)-dimensional Fano manifold. Let \(r\) be the index of \(X\), \(\rho\) the Picard number of \(X\) and \(L_X\) the line bundle such that \(rL \cong -K_X\). Motivated by a conjecture of Mukai the author proves several theorem with as a thesis that \(X\) is a multiprojective space (with factors of prescribed dimensions) and with as assumptions the existence of ``many'' free divisors. For instance, it is proved that if there exists \(\rho\) non-trivial free divisors \(D_1,\dots ,D_\rho\) with \(D_1+\cdots +D_\rho \in |L_X|\), then \(X \cong (\mathbb {P}^{r-1})^\rho\).
scientific article; zbMATH DE number 7604897
scientific article