Related papers: On the Picard number of almost Fano threefolds wit…

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components…

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…

We study congruences of lines $X_\omega$ defined by a sufficiently general choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the…

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high…

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…

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal…

We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show…

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…

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical…

We prove an optimal Kawamata-Miyaoka-type inequality for terminal $\mathbb Q$-Fano threefolds with Fano index at least $3$. As an application, any terminal $\mathbb Q$-Fano threefold $X$ satisfies the following Kawamata-Miyaoka-type…

We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we…

We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how…

A $n$-dimensional Gorenstein toric Fano variety $X$ is called Del Pezzo variety if the anticanonical class $-K_X$ is a $(n-1)$-multiple of a Cartier divisor. Our purpose is to give a complete biregular classfication of Gorenstein toric Del…

We solve the infinitesimal Torelli problem for $3$-dimensional quasi-smooth ${\mathbb{Q}}$-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively…

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove the…

We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin…

A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\lambda_0,...,\lambda_n). We see how the singularities of…

Let $X$ be a minimal projective 3-fold of general type. The pluricanonical section index $\delta(X)$ is defined to be the minimal integer $m$ so that $P_{m}(X)\geq 2$. According to Chen-Chen, one has either $1\leq \delta(X)\leq 15$ or…

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…