On the Picard number of singular Fano varieties (Q2878698)

A normal projective variety \(X\) is called Fano if there exists some \(n \in \mathbb{N}\) such that \(-nK_X\) is an ample Cartier divisor. Let \(N_1(X)\) be the \(\mathbb{R}\)-vector space of 1-cycles on \(X\) with real coefficients modulo numerical equivalence. The dimension of \(N_1(X)\) is called the Picard number of \(X\) and is denoted by \(\rho(X)\). It is known that if \(X\) is smooth of dimension 2, then \(\rho(X)\leq 9\) and if \(X\) is smooth of dimension 3 then \(\rho(X)\leq 10\).NEWLINENEWLINEIn this paper, the author studies the Picard number \(\rho(X)\) of a \(\mathbb{Q}\)-factorial Gorenstein Fano variety of dimension n defined over an algebraically closed field of characteristic zero such that \(X\) has canonical singularities with at most finitely many nonterminal singularities. It is shown that for any prime divisor \(D\subset X\), \(\rho(X)-\rho(D) \leq 8\). This result was proved in the case when \(X\) is a smooth Fano variety by \textit{C. Casagrande} in [Ann. Sci. Éc. Norm. Supér. (4) 45, No. 3, 363--403 (2012; Zbl 1267.14050)]. The method that the author uses to prove this result is heavily based on the one used by C. Cazagrande in the aforementioned paper to prove the smooth case. This inequality could in principle be used to obtain bounds for \(\rho(X)\) by using inductive arguments on the dimension.NEWLINENEWLINEAs an application the author shows that if \(X\) is a three dimensional \(\mathbb{Q}\)-factorial Gorenstein Fano variety defined over an algebraically closed field of characteristic zero such that \(X\) has canonical singularities with at most finitely many nonterminal singularities then \(\rho(X)\leq 10\).