Related papers: Double quadrics with large automorphism groups

It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$…

Let $X$ be a smooth Fano fourfold admitting a conic bundle structure. We show that $X$ is toric if and only if $X$ admits an amplified endomorphism; in this case, $X$ is a rational variety.

Fano varieties are subvarieties of the Grassmannian whose points parametrize linear subspaces contained in a given projective variety. These expository notes give an account of results on Fano varieties of complete intersections, with a…

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying…

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general…

We study nodal del Pezzo 3-folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with…

We initiate the study of finite abelian groups that faithfully act on 3-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: the groups of product type are finite abelian groups…

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only…

We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective space over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an…

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

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete…

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).

The map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order…

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

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…

The anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. In the present article we enlarge…

We determine the automorphism group for a large class of affine quadric hypersurfaces over a field, viewed as affine algebraic varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere is just the…