On the existence of good divisors on Fano varieties of coindex 3

We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.

We start the classification of smooth projective threefolds X whose anticanonical bundles -K_X are big and nef but not ample. In this paper we treat the case b_2(X) = 2 and the morphism associated with the base point free linear system |-mK_X|, m>>0, is divisorial.

We study Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X) = 1, Q-factorial terminal singularities and -K_X = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties X,A and deduce both the nonvanishing of H^0(X,-K_X) and the sharp bound (-K_X)^3 >= 8/165. We list families that can be realised in codimension up to 4.

We show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover,if a general element of the anticanonical system is reduced, then it has canonical singularities. We also prove nonvanishing of anticanonical system in the case of log canonical Gorenstein singularities.

We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.

We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.

Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ nef and not semi-ample. In our previous work, we classified all such threefolds when $|{-}K_X|$ has no fixed divisor. In this paper, we continue our classification when $|{-}K_X|$ has a non-zero fixed divisor.

Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and birationality of the divisor mK_X. For example, we know the following results: (1) the system |5rK_X| is base point free (Ein-Lazarsfeld-Lee); (2) |mK_X| gives a birational map for all m>4r+2 (M. Hanamura). This article aims to present a better r...

Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$ for every curve $C \subset X$, then $\rho_X=1$.

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano...