Geometric Invariant Theory and Enhanced Master Space

We recall how to construct moduli stacks by using the geometric invariant theory. Then, we construct enhanced master spaces in our situation, and we describe the fixed point set with respect to a natural torus action.

In Section 4.1, we review a basic result on the geometric invariant theory for our construction of moduli stacks of δ-semistable parabolic L-Bradlow pairs. In Section 4.2, we consider a perturbation of a δ-semistability condition. Namely, we multiply a full flag bundle to quot schemes, and we argue what is obtained for small perturbation of semistability conditions.

The results in Sections 4.3–4.4 are one of the cores of this paper, which are useful in showing our transition formula. In Section 4.3, we construct our enhanced master space, and we show that it is Deligne-Mumford and proper. In Section 4.4, we study the fixed point set with respect to a natural torus action.

In Section 4.5, we construct an enhanced master space in the oriented case, and we give a description of the stack theoretic fixed point set with respect to the natural torus action. They are essentially just a reformulation of the results in the previous sections.We give a more convenient description of the fixed point set in Section 4.6, i.e., we observe that they are isomorphic to the product of moduli stacks of objects with lower ranks, up to étale proper morphisms.

In some simpler cases, we do not have to consider enhanced master spaces. The statements for such cases are given in Section 4.7.

In this chapter, let X be a smooth connected d-dimensional projective variety over an algebraically closed field k of characteristic 0, and let O_X(1) be a very ample line bundle. Put g := ∫_X c₁(O_X(1))^d. Let D denote a Cartier divisor of X. We do not have to assume smoothness of D in this chapter. We sometimes use the symbol k to denote some numbers. We hope that there are no confusion.