Toric P-difference varieties

In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebraic geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine P [x]-semimodules with P [x]-semimodule morphisms. Moreover, there is a one-to-one correspondence between the irreducible invariant P-difference subvarieties of an affine toric P-difference variety and the faces of the corresponding affine P [x]-semimodule. We also define abstract toric P-difference varieties by gluing affine toric P-difference varieties. The irreducible invariant P-difference subvariety-face correspondence is generalized to abstract toric P-difference varieties. By virtue of this correspondence, a divisor theory for abstract toric P-difference varieties is developed.     

Extended Deligne–Lusztig varieties for general and special linear groups

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group.Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL₂(Fq???/(?²)), with odd q, the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.     

Degenerate Affine Flag Varieties and Quiver Grassmannians

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

     

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

Order varieties generated by finite posets

In a 1981 paper, Duffus and Rival define an order variety as a class of posets that is closed under the formation of products and retracts. They also introduce the notion of an irreducible poset. In the present paper we define nonextendible colored posets and certain minimal nonextendible colored posets that we call zigzags. We characterize via nonextendible colored posets the order varieties generated by a set of posets. If the generating set contains only finite posets our characterization is via zigzags. By using these theorems we give a characterization of finite irreducible posets.As an application we show that two different finite irreducible posets generate two different order varieties. We also show that there is a poset which has two different representations by irreducible posets. We thereby settle two open problems listed in the Duffus and Rival paper.     

Varieties of rings,where all subdirectly irreducible finite rings are Armendariz ones

In this paper we study varieties of rings, where all subdirectly irreducible finite rings are Armendariz. We also describe the locally finite varieties of Armendariz rings.     

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.     

Decompositions of determinantal varieties

In this paper, we study the irreducible decompositions of determinantal varieties of matrices given by rank conditions on upper left submatrices. Using the concept of essential rank function and the Ehresmann partial order on the set of all simple matrices, we design an algorithm to write a determinantal variety as a union of its irreducible components. This solves a problem raised by B. Sturmfels.      

Crystal bases and quiver varieties

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

Degenerations of binary Lie and nilpotent Malcev algebras

We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over ?. In particular, we describe all irreducible components of these varieties.     

State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.     

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.     

Mixed ladder determinantal varieties from two-sided ladders

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay, and we characterize the arithmetically Gorenstein ones. Our main result consists in proving that mixed ladder determinantal varieties belong to the same G-biliaison class of a linear variety.     

Refined Noether Normalization Theorem and Sharp Degree Bounds for Dominating Morphisms

Abstract

After the proof of a not very known refinement of the Noether Normalization Theorem, we obtain two sharp degree bounds for the geometric degree of a dominating morphism of irreducible affine algebraic varieties and for the degree of the components of the inverse of an isomorphism of such varieties, generalizing by this way the well-known Gabber bound for automorphisms of affine spaces.     

Mustafin varieties

A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the 2-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.