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This paper is a summary of author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed…

Using Galois descent tools, we extend the Altmann-Hausen presentation of normal affine algebraic varieties endowed with an effective torus action over an algebraically closed field of characteristic zero to the case where the ground field…

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie…

In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of…

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for…

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can…

We identify all smooth manifolds of curves for Heath-Jarrow-Morton models that are consistent with any tangential diffusion coefficient. In fact, we show that these manifolds cannot be affine but must be of linear-rational type.

The paper concerns the affine varieties that are homogeneous with respect to a (non-standard) graduation over the group Z^m. Among the other properties it is shown that every such a variety can be embedded in its Zariski tangent space at…

In this paper, we will show that the affine cones over any smooth Fano-Mukai fourfold of genus $7$ are flexible.

In this paper, we study a local rigidity property of $\mathbb Z \ltimes_\lambda \mathbb R$ affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of…

Let $X$ be a factorial complex affine variety of dimension $\ge 3$ with an algebraic action of the additive group $G_a$. Let $\pi : X \to Y$ be the algebraic quotient morphism where we assume $Y$ is an affine variety. When $\pi$ is…

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

Let Z be an affine algebraic variety and ED(Z)= max(2 dim Z+1, dim TZ). Let X be a smooth algebraic variety isomorphic to a semi-simple linear algebraic group whose Lie algebra is a sum of special linear Lie algebras. We show that if dim X…

In this paper we show that a normal affine toric variety X different from the algebraic torus is uniquely determined by its automorphism group in the category of affine irreducible, not necessarily normal, algebraic varieties if and only if…

We establish a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties, that is, toric fibrations over rational homogeneous spaces. The key ingredient is a characterization of the Demazure…

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the…

Affine Lusztig varieties encode the orbital integrals of Iwahori--Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. We establish a close relationship between affine Lusztig varieties and…

In this paper we investigate aspects of rigidity and flexibility for conformal iterated function systems. For the case in which the systems are not essentially affine we show that two such systems are conformal equivalent if and only if in…

We show that for two afii varieties over an arbitrary field of characteristic zero, there is no general form of an algorithm for checking the presence of an embedding of one algebraic variety in another. Moreover, we establish this for…