Related papers: Gorenstein Fano threefolds with base points in the…

For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $\varphi_{-m}$ is birational onto its…

We give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. As a consequence we characterize Fano Bott manifolds.

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau…

We classify birationally rigid orbifold Fano 3-folds of index one defined by $5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various…

We classify Q-Fano threefolds of Fano index > 2 and big degree.

We obtain a classification of a Q-factorial Q-Fano 3-fold $X$ with the following properties: the Picard number of $X$ is 1; the Gorenstein index of $X$ is 2; the Fano index of $X$ is 1/2; $h^0 (-K_X) \geq 4$; there exists an index 2 point…

In this paper, an update on the classification of smooth weak Fano threefolds with Picard number two and small anti-canonical maps is given. Geometric constructions are provided for previously open numerical cases by blowing up certain…

We classify completely the surfaces of general type whose canonical map is 3-to-1 onto a surface of minimal degree in projective space. These surfaces fall into 5 distinct classes and we give explicit examples belonging to each of these…

We study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities…

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein…

We determine the fixed locus of the anticanonical complete linear system of a given anticanonical rational surface. The case of a geometrically ruled rational surface is fully studied, e.g., the monoid of numerically effective divisor…

We study three-dimensional Fano varieties with $\mathbb{C}^*$-action. Complementing recent results [13], we give classification results in the canonical case, where the maximal orbit quotient is $\mathbb{P}_2$ having a line arrangement of…

In this paper, we give a characterization of Fano type varieties in terms of the asymptotic base loci of $-(K_X+\Delta)$. We also show that for a potentially lc pair $(X,\Delta)$, if no plc centers are contained in the augmented base locus…

We prove that the Albanese map of a smooth projective threefold, whose anticanonical bundle is nef, is a surjective submersion. We also investigate morphisms of threefolds to curves and surfaces whose relative anticanonical bundle are nef.

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For…

For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.

We prove that every smooth Fano threefold from the family No 2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of $\mathbb{P}^3$ at one point branched along an anti-canonical divisor.

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the K\"ahler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20,…

We give a characterizaion of Gorenstein toric Fano $n$-folds with index $n-1$, which is called Gorenstein toric Del Pezzo $n$-folds, among toric varieties. In practice, we obtain a condition for a lattice $n$-polytope to be a Gorenstein…