Related papers: Equivariant compactifications of vector groups wit…
In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_a^3$ which are smooth Fano threefolds with Picard number greater or equal than two.
In this paper we classify n-dimensional Fano manifolds with index >=n-2 and positive second Chern character.
Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many…
Let $X$ be a complex smooth Fano variety of dimension at least four. In this paper, we classify such $X$ when the pseudoindex is at least $n-2$ and the Picard number greater than one. We also discuss the relations between pseudoindex and…
For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a…
We classify Fano fivefolds of index two which are blow-ups of smooth manifolds along a smooth center.
We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian variety in dimension 8 is in this family. The 8-fold has interesting properties:…
We classify Fano fivefolds of index two which are projectivization of rank two vector bundles over four dimensional manifolds.
Equivariant compactifications of reductive groups can be described by combinatorial data. On the other hand, equivariant compactifications of the additive group G^n_a are more complicated in at least two respects. First, they often admit…
We classify smooth Fano manifolds X with the Picard number $\rho_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.
In the present paper we discuss coherent sheaves of rank > 1 whose projectivization gives rise to smooth varieties - varieties of this type are also called smooth scrolls. We prove some basic properties of these varieties and we give some…
For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\bar{G}$, which is a smooth Fano $G \times G$-variety of Picard number $n$…
A normal projective variety X is called Fano if a multiple of the anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier divisor H}. Mukai…
In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.
We classify smooth Fano threefolds with infinite automorphism groups.
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this…
We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the…
Let $\mathcal{F}(n)$ be the set of smooth Fano $n$-polytopes up to unimocular equivalence. In this paper, we consider the F-equivalence or I-equivalence classes for $\mathcal{F}(n)$ and introduce F-isolated or I-isolated smooth Fano…
We classify Q-Fano threefolds of Fano index > 2 and big degree.
We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with…