Moduli of Admissible Pairs for Arbitrary Dimension, II: Functors and Moduli

Admissible pairs $((\tilde{S},\tilde{L}),\tilde{E})$ consisting of an $N$-dimensional projective scheme~$\tilde{S}$ of certain class with a special ample invertible sheaf $\tilde{L}$ and a locally free ${\cal O}_{\widetilde S}$-sheaf$\widetilde E$ are considered. An admissible pair can be produced in the procedure of a transformation (which is called a resolution) of a torsion-free coherent sheaf $E$ on a nonsingular $N$-dimensional projective algebraic variety $S$ to a locally free sheaf $\tilde{E}$ on some projective scheme $\tilde{S}$. Notions of stability and semistability of admissible pairs and a moduli functor for semistable admissible pairs are introduced. Also the relation of the stability and semistability for admissible pairs to the classical stability and semistability for coherent sheaves under the resolution is examined. Morphisms between the moduli functor of admissible semistable pairs and the Gieseker–Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on a nonsingular $N$-dimensional projective algebraic variety are constructed. Examining these morphisms it is shown that the moduli scheme for semistable admissible pairs $((\tilde{S},\tilde{L}),\tilde{E})$ is isomorphic to the Gieseker–Maruyama moduli scheme for coherent sheaves. The considerations involve all the existing components of these moduli functors and of their corresponding moduli schemes. Bibliography: 25 items.