A functorial construction of moduli of sheaves
This paper surveys GIT (geometric invariant theory) with an additional new insight coming from the presentation of coherent sheaves as Kronecker modules over an associative algebra. These Kronecker modules are modules for a path algebra of an easy quiver. This allows to give explicit theta functions on the eventually obtained moduli space. This paper is a detailed introduction to the construction of moduli spaces of coherent sheaves over a fixed projective scheme. To describe the result of the article, let us fix notation: \noindent \(X\), \(\mathcal O(1)\) is a projective scheme over an algebraically closed field \(k\), \(m>n\) are two integers (subject to some conditions), \(T\) is the sheaf \(\mathcal O(-n) \oplus \mathcal O(-m)\), and \(A\) is the endomorphism algebra End\((T)\). In section 3 they show that Castelnuovo-Mumford regularity can be rephrased by saying that for a \(n\)-regular sheaf \(E\) the evaluation morphism Hom\((T,E) \otimes _A T \to E\) is an isomorphism. This implies that the embedding functor \(E \mapsto \) Hom\((T,E)\) is an embedding and \(E\) can be reconstructed from its image, whenever \(m \gg n\). In section 4 the embedding functor is applied to families. The key ingredient for this new presentation of GIT is Theorem 5.10 where for pure and \(n\)-regular sheaves the preservation of stability/semistability is shown. In section 6 they present the construction of the moduli space of semistable sheaves with given Hilbert polynomial on \(X\) using the map to the moduli of Kronecker modules over \(A\). \noindent They show that this morphism \(\mathcal M_X^{ss} \to \mathcal M_A^{ss}\) is a closed embedding when char\((k)=0\). For char\((k)>0\) the authors show that it is a closed embedding on the stable locus. In the final section 7 they use results of the representation theory of quivers to define theta function on the moduli space, and investigate their separation properties. The global generatedness of these theta functions gives a nice semistability criterion for \(n\)-regular, pure sheaves on \(X\) (see Theorem 7.2) which says that \(E\) is semistable if a complex \(\delta: \mathcal O_X(-m)^{\oplus u_1} \to \mathcal O_X(-n)^{\oplus u_0}\) is cohomologically perpendicular to \(E\).
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