Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.     

Fano manifolds obtained by blowing up along curves with maximal Picard number

The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. In this note, we show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a hyperquadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.     

Moduli of reflexive sheaves on smooth projective 3-folds

We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf, and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.     

A remark on Fano 4-folds having (3,1)-type extremal contractions

Let π : X → Y be the blow-up of a four-dimensional complex manifold Y along a smooth curve C. Assume that X is a Fano manifold and has another (3,1)-type extremal contraction ${\varphi : X \to Z}$ whose exceptional divisor meet that of the blow-up π : X → Y. We show that if the exceptional divisor of ${\varphi}$ is smooth, then Y is isomorphic to four-dimensional projective space and C is an elliptic curve of degree 4.     

Three embeddings of the Klein simple group into the Cremona group of rank three

We study the action of the Klein simple group PSL₂( $ {\mathbb{F}_7} $ ) consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL₂( $ {\mathbb{F}_7} $ ). As a by-product, we prove that X admits a K?hler?CEinstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group PSL₂( $ {\mathbb{F}_7} $ ). Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.     

Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.     

Low-dimensional projective manifolds with nef tangent bundle in positive characteristic

We classify smooth projective varieties with nef tangent bundle in positive characteristic, when the varieties are surfaces or Fano 3-folds. Furthermore, some related problems will be discussed.     

Upper bounds for errors of estimators in a problem of nonparametric regression: the adaptive case and the case of unknown measure ρX

The topological type of the real part of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is evaluated provided that the hypersurface belongs to a special rigid projective class. In the paper by Finashin and Kharlamov on the rigid projective classification of real four-dimensional cubics, this class is said to be irregular. The results of the author of the present paper from the article devoted to the equivariant topological classification of the Fano varieties of real cubic fourfolds are also used.     

Flat deformations of ℙ n

In this paper we study projective flat deformations of ? ⁿ . We prove that the singular fibers of a projective flat deformation of ? ⁿ appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of ? ⁿ with smooth total space, and discuss flatness criteria.     

Fano subplanes in finite Figueroa planes

It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. The Figueroa planes are a family of non-translation planes that are defined for both infinite orders and finite order q ³ for q > 2 a prime power. We will show that there is an embedded Fano subplane in the Figueroa plane of order q ³ for q any prime power.     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Polarized Calabi–Yau 3‐folds in codimension 4

We construct Calabi–Yau 3‐fold orbifolds embedded in weighted projective space in codimension 4. Each Hilbert series we consider is realised by at least two deformation families of Calabi–Yau 3‐folds, distinguished by their topology, echoing a similar phenomenon for Fano 3‐folds in high codimension.     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of A. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ.     

Unramified Brauer group of the moduli spaces of PGL r (ℂ)-bundles over curves

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGL _r (?)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.     

Derived categories of Fano threefolds

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 116–128. In memory of V.A. Iskovskikh     

Existence of natural and projectively equivariant quantizations

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operators, and prove the existence of a quantization except when this parameter belongs to a discrete set of resonant values.