We prove that the sum of the αα-invariants of two different Kollár components of a Kawamata log terminal singularity is less than 1.

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group *G*, which are not *G*-rational and have no birational structure of *G*-Mori fiber space with the base of positive dimension. Also we prove the 𝔄5A5-birational superrigidity of the Segre cubic.

We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.

According to the classical theorem, every algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and the projective space of a positive dimension. We show that the classical proof of this theorem actually works only in characteristic 0 and we give a characteristic free proof of it. To this end we prove and use a characterization of connected linear algebraic groups G with the property that every rational action of G on an irreducible algebraic variety is birationally equivalent to a regular action of G on an affine algebraic variety.

This work was done during authors’ visit to Kavli IPMU and was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The reported study was partially supported by RFBR, Research Projects 13-01-00234, 14-01-00416 and 15-51-50045. The article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.

For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over deRham DG-algebras. Positselski proved that for a smooth algebraic variety X over a field k of characteristic zero the coderived category of DGmodules over •X/k is equivalent to the unbounded derived category of quasi-coherent right DX -modules.We prove that our functors correspond to the functors of the same name for DX -modules under Positselski equivalence.

Given two elliptic curves, each of which is associated with a projection map that identifies opposite elements with respect to the natural group structure, we investigate how their corresponding projective images of torsion points intersect.

We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.

Abstract We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for a regular behavior of the equiclassical stratification. We also discuss duality of curves in higherdimensional projective spaces and in Grassmannians with focus on similar questions of the regularity of equiclassical families of spacial curves.

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z ×A1, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto et al.

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove that ρw(X)+1 pairwise-disjoint, connected divisors suffice for proper, normal varieties X, where ρw(X) is a modification of the Néron–Severi rank of X (they agree when X is projective and smooth). We then prove a strong counterexample in the affine case: if X is quasi-affine and of dimension ⩾ 2 over a countable, algebraically-closed field k, then there exists a (countable) collection of pairwise-disjoint divisors which cover the k-points of X, so that for any non-constant morphism from X to a curve, at most finitely many are contained in the fibers thereof. We show, however, that an uncountable collection of pairwise-disjoint, connected divisors in any normal variety over an algebraically-closed field must be contained in the fibers of a morphism to a curve.

It is known that the moduli space of smooth Fano–Mukai fourfolds *V*18 of genus 10 has dimension one. We show that any such fourfold is a completion of ℂ4 in two different ways. Up to isomorphism, there is a unique fourfold *V*s18 acted upon by SL2(ℂ). The group Open image in new window is a semidirect product Open image in new window . Furthermore, *V*s18 is a GL2(ℂ)-equivariant completion of ℂ4, and as well of GL2(ℂ). The restriction of the GL2(ℂ)-action on *V*s18 to Open image in new window yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold *V*a18 such that the group Open image in new window is a semidirect product Open image in new window . For a Fano–Mukai fourfold *V*18 isomorphic neither to *V*s18, nor to *V*a18, the group Open image in new window is a semidirect product of (𝔾m)2 and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety *V*18 is flexible in codimension one, and flexible if *V*18=*V*s18.

We study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.

We develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra.

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function

$\Delta(\tau)=\eta^{24}(\tau)$, $\eta^{12}(\tau)$ and $\eta^{8}(\tau)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.

We study quartic double solids admitting icosahedral symmetry.

Let 𝕜 be any field of characteristic zero, X be a cubic surface in ℙ3 and G be a group acting on X. We show that if X(𝕜)≠∅ and G is not trivial and not a group of order 3 acting in a special way then the quotient surface X/G is rational over 𝕜. For the group G of order 3 we construct examples of both rational and nonrational quotients of both rational and nonrational G-minimal cubic surfaces over 𝕜.