The Moduli Space of (3,3,3) Trilinear forms.
The moduli space of totally marked degree two rational maps
A rational map ϕ: ℙ¹ → ℙ¹ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps can be parametrized by an affine open subset of (ℙ¹)⁵. We consider the natural action of SL₂ on induced from the action of SL₂ on (ℙ¹)⁵ and prove that the quotient space exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over ℤ[1/2].
The Mordell–Lang question for endomorphisms of semiabelian varieties
The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...
The multipliers of similitudes and the Brauer group of homogeneous varieties.
The Mumford-Tate group of 1-motives
In this paper we study the structure and the degeneracies of the Mumford-Tate group of a 1-motive defined over . This group is an algebraic - group acting on the Hodge realization of and endowed with an increasing filtration . We prove that the unipotent radical of , which is , injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1....
The normality of closures of conjugacy classes of matrices.
The normality of closures of orbits in a Lie algebra.
The number of points on certain families of hypersurfaces over finite fields
The Number of Rational Quartics on Calabi-Yau Hypersurfaces in Weighted Projective Space P (2,1...).
The Picard Group of a complex quotient variety.
The Picard-Vessiot closure in differential Galois theory
The quasi-hereditary algebra associated to the radical bimodule over a hereditary algebra
Let Γ be a finite-dimensional hereditary basic algebra. We consider the radical rad Γ as a Γ-bimodule. It is known that there exists a quasi-hereditary algebra 𝓐 such that the category of matrices over rad Γ is equivalent to the category of Δ-filtered 𝓐-modules ℱ(𝓐,Δ). In this note we determine the quasi-hereditary algebra 𝓐 and prove certain properties of its module category.
The scheme of connected components of the Néron model of an algebraic torus.
The split building of a reductive group.
The strong linkage principle.
The structure of proper rigid groups.
The trace formula for equivariant D-modules and perverse sheaves.
The Two Most Algebraic K3 Surfaces.
The variety of dual mock-Lie algebras
We classify all complex - and -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex -dimensional dual mock-Lie algebras.
The Waring loci of ternary quartics.