On a new compactification of moduli of vector bundles on a surface, IV: Nonreduced moduli

The construction for nonreduced projective moduli scheme of semistable admissible pairs is performed. We establish the relation of this moduli scheme with reduced moduli scheme built up in the previous article and prove that nonreduced moduli scheme contains an open subscheme which is isomorphic to moduli scheme of semistable vector bundles.Comment: 20 pages, additions and removal

Fibred product of commutative algebras: generators and relations

The method of direct computation of universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces the dimension of their product obeys Grassmann-like formula. Finite-dimensional case means geometrically the strict version of adding two collections of points containing some common part. The method involves description of algebras by generators and relations on input and returns similar description of the product algebra. It is "ready-to-eat" even for computer realization. The product algebra is well-defined: taking another descriptions of the same algebras leads to isomorphic product algebra. Also it is proven that the product algebra enjoys universal property, i.e. it is indeed fibred product. The input data is a triple of algebras and a pair of homomorphisms $A_1\stackrel{f_1}{\to}A_0 \stackrel{f_2}{\leftarrow}A_2$. Algebras and homomorphisms can be described in any fashion. We prove that for computing the fibred product it is enough to restrict to the case when $f_i,i=1,2,$ are surjective and describe how to reduce to surjective case. Also the way to choose generators and relations for input algebras is considered.Comment: 15 pages, generalization of result of previous versio

Formation of the human resource capacity in modern universities

The paper discussed the personnel issue, its importance and problems of its formation in higher educational institutions

On a new compactification of moduli of vector bundles on a surface, III: Functorial approach

A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial, on the smooth projective polarized surface (S,L), is constructed. Families of locally free sheaves on the surface S are completed by locally free sheaves on schemes which are modifications of S. Gieseker -- Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs polarized scheme -- vector bundle with moduli space of such type.Comment: 53 pages, typo in sec.12 correcte