## 14D20 Algebraic moduli problems, moduli of vector bundles (For analytic moduli problems, see 32G13)

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This thesis deals with the following question. Given a moduli space of coherent sheaves on a projective variety with a fixed Hilbert polynomial, to find a natural construction that replaces the subvariety of the sheaves that are not locally free on their support (we call such sheaves singular) by some variety consisting of sheaves that are locally free on their support. We consider this problem on the example of the coherent sheaves on \(\mathbb P_2\) with Hilbert polynomial 3m+1.
Given a singular coherent sheaf \(\mathcal F\) with singular curve C as its support we replace \(\mathcal F\) by locally free sheaves \(\mathcal E\) supported on a reducible curve \(C_0\cup C_1\), where \(C_0\) is a partial normalization of C and \(C_1\) is an extra curve bearing the degree of \(\mathcal E\). These bundles resemble the bundles considered by Nagaraj and Seshadri. Many properties of the singular 3m+1 sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles). We consider R-bundles as natural replacements of the singular sheaves. R-bundles refine the information about 3m+1 sheaves on \(\mathbb P_2\). Namely, for every isomorphism class of singular 3m+1 sheaves there are \(\mathbb P_1\) many equivalence classes of R-bundles. There is a variety \(\tilde M\) of dimension 10 that may be considered as the space of all the isomorphism classes of the non-singular 3m+1 sheaves on \(\mathbb P_2\) together with all the equivalence classes of all R-bundles. This variety is obtained by blowing up the moduli space of 3m+1 sheaves on \(\mathbb P_2\) along the subvariety of singular sheaves. We modify the definition of a 3m+1 family and obtain a notion of a new family over an arbitrary variety S. In particular 3m+1 families of the non-singular sheaves on \(\mathbb P_2\) are families in this sense. New families over one point are either non-singular 3m+1 sheaves or R-bundles. For every variety S we introduce an equivalence relation on the set of all new families over S. The notion of equivalence for families over one point coincides with isomorphism for non-singular 3m+1 sheaves and with equivalence for R-bundles. We obtain a moduli functor \(\tilde{\mathcal M}:(Sch) \rightarrow (Sets)\) that assigns to every variety S the set of the equivalence classes of the new families over S. There is a natural transformation of functors \(\tilde{\mathcal M}\rightarrow \mathcal M\) that establishes a relation between \(\tilde{\mathcal M}\) and the moduli functor \(\mathcal M\) of the 3m+1 moduli problem on \(\mathbb P_2\). There is also a natural transformation \(\tilde{\mathcal M} \rightarrow Hom(\__ ,\tilde M)\), inducing a bijection \(\tilde{\mathcal M}(pt)\cong \tilde M\), which means that \(\tilde M\) is a coarse moduli space of the moduli problem \(\tilde{\mathcal M}\).

In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.

We extend the methods of geometric invariant theory to actions of non reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non recutive. Given a linearization of the natural actionof the group Aut(E)xAut(F) on Hom(E,F), a homomorphism iscalled stable if its orbit with respect to the unipotentradical is contained in the stable locus with respect to thenatural reductive subgroup of the automorphism group. Weencounter effective numerical conditions for a linearizationsuch that the corresponding open set of semi-stable homomorphismsadmits a good and projective quotient in the sense of geometricinvariant theory, and that this quotient is in additiona geometric quotient on the set of stable homomorphisms.