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

We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group $G$. In particular, we find a lot of examples of Fano 3-folds with "many" symmetries.

We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum of Fano index is attained for the weighted projective space P(3,4,5,7).

In this article we prove some strong vanishing theorems on K3 surfaces. As an aplication of them, we obtain higher syzygy results for K3 surfaces and Fano varieties.

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+\omega x_1^2+\omega^2x_2^2+\big(x_3^2+\omega x_4^2+\omega^2x_5^2\big)+\big(x_0x_3+\omega x_1x_4+\omega^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=...

Following the first paper, we continue to study Mori extractions from singular curves centred in a smooth 3-fold. We treat the case where the divisorial extraction exists in relative codimension at most 3.

We show that there exists a K-stable smooth Fano threefold of the Picard rank 3, the anti-canonical degree 28 and the third Betti number 2.

We consider Fano threefolds $V$ with canonical Gorenstein singularities. A sharp bound $-K_V^3\le 72$ of the degree is proved.

Let $(X,D)$ be log canonical pair such $\dim X = 3$ and the divisor $-(K_X + D)$ is nef and big. For a special class of such $(X,D)$'s we prove that the linear system $|-n(K_{X}+D)|$ is free for $n \gg 0$.