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A new algebraic method for the investigation of vectorbundles on the projective plane

Classical terms and summary

In this work we place a new algebraic method to the investigation of vector bundles, in addition, arbitrary coherent modules on projective spaces ℙn. To motivate the introduction of the essential term, the Kronecker-module, connected with coherent ℙn-modules, we want to remind now of the classical results of Serre.

Let Gradn be the category of finitely generated modules over the graduated k-algebra of the polynomials, k[x0....xn], whereby k is an algebraically closed field. From a module M in Gradn we can construe a ℙn-module F in the following way: We define the cuts Γ(ℙnf,F) of F over the open set ℙnf of ℙn of the points, where the homogeneous function f∈k[x1,...,x n] does not disappear, as the 0-component (M f)0 of the localization Mf of M concerning the set {1, f, f2,f3...}. Thereby is determined a coherent ℙn-module F up to single isomorphism.

In particular k[x0...,xn](m), received from k[x0...,xn] by shifting the graduation, is assigned to the locally free module ℙn-module O(m), a line bundle, with m<0 a power of the Hopf-bundle O(-1). If we name F(m) the tensor-product FO(m) and Γ(F) the global cuts of F, then for a coherent ℙn-module F the vector space X(F) = ⊕Γ(F(m)) (direct sum) is a finitely generated k[x0,...,xn]-module, provided with the natural multiplication, which is defined as follows: We understand xi as cut of O(1) and define xi⋅s for s in Γ(F(m)) as the cut of O(1)⊗F(m) = F(m+1).

In this way we receive two functors A: GradnMod and X: Mod→Gradn, whereby Mod designates the category of the coherent ℙn-module-sheaves. The construction A of ℙn-modules does not depend however on the homogeneous components on the degrees smaller than some arbitrary constant, and on the other hand there are modules with X(F)m = 0 for all  m∈ℤ  smaller than some arbitrary constant. Therefore the functors A and X are only then "inverse" if we in Gradn identify objects and morphisms, which agree in all homogeneous components of sufficient high degree.

This identification in Gradn is however purely theoretical and practically one works with the objects in Gradn. But then it can occur that e.g. an irreducible graduated module M induces a reducible ℙn-module A(M). In order to loose-come however from the necessity for the class formation, one must itself in each case limit on ℙn-modules F, which already are constructable from its cuts Γ(F(m)) for  m  smaller than a constant. For example the subcategorie Vb0n of Modn of the locally free globally generated ℙn-moduls F, where the higher cohomologie of the F(m) for m m≥-n  disappears, posseses this characteristic, and we will show that such a module is already determined by two components of X(F), e.g. by Γ(F(-1)) and Γ(F), together with the multiplication of the xi∈k[x0,...,xn], which induce n+1 linear mappings xi: Γ(F(-1))→Γ(F). Such an object of two vector spaces V and W with a finite number k of linear mappings V→W is called Kronecker-module with k arrows.

We can thus assign to each coherent ℙn-module F a Kronecker-module κ(F) with n+1 arrows Γ(F(-1))→Γ(F). We develop also a procedure for the global construction of ℙn-modules M(æ) from Kronecker-modules  æ  and show that the functor  κ  on a subcategorie Mod 0n of Mod, who contains Vb0n, induce a fully faithful embedding into the category of the Kronecker-modules and is a "cut" for the M. The category Mod 0n has the characteristic that there exists for each coherent ℙn-module F an m∈ℕ, so that F(m) lies in Mod0 n. Therefore one can limit oneself to Mod0n, if one regards ℙn-modules only up to isomorphism, since F(m) is isomorphic to G (m) exactly if F isomorphic to G.

Since we are interested in vector bundles primarily by questions of isomorphism, we examine characteristics of the Kronecker-modules of vector bundles in Mod 0 n in this work. First however we give a memory of the "classical" methods for designing a quasiprojektives scheme of isomorphism classes of vector bundles over a non-singular projective curve X. One designs first a family or a scheme of vector bundles with a reductive group operating on it in such a way that its orbits are exactly the isomorphism classes. Then one studies the group operation and marks the open subscheme of the "stable" points, for which a "geometrical" quotient, called moduli-scheme, of the group operation by the moduli-group exists, i.e. essentially: a universal quasiprojektive scheme, whose points parametrize the isomorphism classes. [ D. Mumford, geometric invariantl theory, 0.4 ]

Three substantial characteristics of bundles are used on projektive curves X:
1. The vector bundles are exactly the torsion-free X-modules.
2. Each bundle of F has a filtration 0 = F0 → F1 → ... → Fr-1 → Fr = F, so that Fiis/F i-1 a line bundle.
3. Given the discrete invariants rank r and degree C there is a scheme of bundles of rank r and degree C with a moduli-group.

We are in the situation, to mark the stable ("particularly stable") vector bundles with similar methods on ℙ2 (for arbitrary ranks). Unfortunately a geometrical interpretation of a criterion for stability succeeds only for special (monomial) bundles. The methods used here are surely "developable", but in contrast not for arbitrary bundles over ℙn, because the three central characteristics used for to the derivation of the condition of stability on ℙn reliably do not apply for n > 2.

The following correspond to the above statements in this work:
1. The Kronecker-modules of vector bundles are exactly the regular Kronecker-modules. (for the definition of "regular" the term of torsion-freedom is crucial. In addition it is clear that the Kronecker-module of a vector bundle is torsion-free, i.e. for e‡0 in Γ(O(1)) and x‡0 in Γ(F(-1)) is e⋅x∈Γ(F) not zero. This applies to all ℙn.
2. Each bundle on ℙ2 has a series 0 = F-1 , F0 ..., Fm-1 , Fm = F of "derived" bundles with exact sequences 0→O⊗Yi→Fi-1(1)→FiO⊗Zi→0, whereby Yi and Zi are vector spaces. DimZi is the maximum rank of the trivial component of F and dimYi can obviously be expressed by ranks of Fi and Fi-1. The proof of the existence unfortunately does not show, how far the validity of this statement is linked with the regularity of κ(F(-1)): E⊗Γ(F(-2))→Γ(F(-1)), where E = Γ(O(1)), and that it cannot apply for ℙn for n > 2, e.g. is κ(Ω(3)) is not regular for n = 3.
3. With default of the invariants m (the "stage" of F) as well as dimYi and dimZi is there a scheme of vector bundles with a moduli-group, here a SLk. The definition is only possible for all bundles on ℙ2, since point 2. applies only to all bundles under the condition n = 2.

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