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Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G0 is a torus and p does not divide the index (G : G0). [ABSTRACT FROM AUTHOR]

Given two affine algebraic varieties X and Y such that X ×  ≅ Y × , it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1. [ABSTRACT FROM AUTHOR]

The article provides information on the inequalities of lattice polygons in relation to Scott's inequality and the equation 2i + 7. It states that algebraic geometry is used to study lattice equivalent polygons, since toric geometry is a powerful link that connects discrete and algebraic geometry. Moreover, the proposition b ⩽ 2i + 7 is proven using Pick's Theorem with three equivalent inequalities. Furthermore, Scott's proof indicates that (x,y) = (3,3) is the only point where both upper bounds reach zero and the only case where equality can hold in Onion-Skin Theorem.

We prove that a finite-dimensional algebra over an algebraically closed field is of derived dimension 0 if and only if it is an iterated tilted algebra of Dynkin type. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

Let X = Spec ( R ) be a reduced equidimensional algebraic variety over an algebraically closed field k . Let Y = Spec ( R /??) be a codimension one ordinary multiple subvariety, where ?? is a prime ideal of height 1 of R . If U is a nonempty open subset of Y and ?? a closed point of U , we denote by A ? R ?? its local ring in X , by ?? the extension of ?? in A , and by K the algebraic closure of the residue field k (??). Then there exists a bijection ? ?? : Proj ( G ?? ( A ) ? A /?? k ) ? Proj ( G ( A ?? ) ? k (?? ) K ) such that for every subset S of Proj ( G ?? ( A ) ? A /?? k ), the Hilbert function of S coincides with the Hilbert function of ? ?? (S). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety. [ABSTRACT FROM AUTHOR]