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 ℙ_n. To motivate the introduction of the essential term, the Kronecker-module, connected with coherent ℙ_n-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....x_n], whereby k is an algebraically closed field. From a module M in Gradⁿ we can construe a ℙ_n-module F in the following way: We define the cuts Γ(ℙ_nf,F) of F over the open set ℙ_nf of ℙ_n 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 M_f of M concerning the set {1, f, f²,f³...}. Thereby is determined a coherent ℙ_n-module F up to single isomorphism.

In particular k[x₀...,x_n](m), received from k[x₀...,x_n] by shifting the graduation, is assigned to the locally free module ℙ_n-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 F⊗O(m) and Γ(F) the global cuts of F, then for a coherent ℙ_n-module F the vector space X(F) = ⊕Γ(F(m)) (direct sum) is a finitely generated k[x₀,...,x_n]-module, provided with the natural multiplication, which is defined as follows: We understand x_i as cut of O(1) and define x_i⋅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: Gradⁿ→Mod and X: Mod→Gradⁿ, whereby Mod designates the category of the coherent ℙ_n-module-sheaves. The construction A of ℙ_n-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 ℙ_n-module A(M). In order to loose-come however from the necessity for the class formation, one must itself in each case limit on ℙ_n-modules F, which already are constructable from its cuts Γ(F(m)) for  m  smaller than a constant. For example the subcategorie Vb₀ⁿ of Mod_n of the locally free globally generated ℙ_n-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 x_i∈k[x₀,...,x_n], which induce n+1 linear mappings x_i: 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 ℙ_n-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 ℙ_n-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 ℙ_n-module F an m∈ℕ, so that F(m) lies in Mod₀ ⁿ. Therefore one can limit oneself to Mod₀ⁿ, if one regards ℙ_n-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 → ... → F_r-1 → F_r = F, so that F_iis/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 ℙ_n, because the three central characteristics used for to the derivation of the condition of stability on ℙ_n 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 ℙ_n.
2. Each bundle on ℙ₂ has a series 0 = F_-1 , F₀ ..., F_m-1 , F_m = F of "derived" bundles with exact sequences 0→O⊗Y_i→F_i-1(1)→F_i→O⊗Z_i→0, whereby Y_i and Z_i are vector spaces. DimZ_i is the maximum rank of the trivial component of F and dimY_i can obviously be expressed by ranks of F_i and F_i-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 ℙ_n 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 dimY_i and dimZ_i is there a scheme of vector bundles with a moduli-group, here a SL_k. The definition is only possible for all bundles on ℙ₂, since point 2. applies only to all bundles under the condition n = 2.