# Parabolic L-Bradlow Pairs

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In this chapter, we recall some basic definitions. All of them are more or less standard. Our purpose is to fix the meanings in this monograph. In the following of this chapter, X will denote a smooth connected projective variety over an algebraically closed field *k* of characteristic 0. Let Pic_{X} denote the Picard variety of X. We fix a base point *x* _{0} Є X, and hence we have a Poincaré bundle *Poin* _{X} on Pic_{X} × X.

In Section 3.1, we review the basic notion. In Subsections 3.1.1–3.1.3, we recall the definition of some structure on torsion-free sheaves such as orientation, parabolic structure, L-section, and reduced L-section. In Subsection 3.1.4, we prepare the symbols to describe some moduli stacks. In Subsection 3.1.5, we introduce relative tautological line bundles of moduli stacks of oriented reduced LBradlow pairs. We also see the relation among moduli stacks of oriented reduced L-Bradlow pairs and unoriented unreduced L-Bradlow pairs.

In Section 3.2, we recall the definition of Hilbert polynomials for torsion-free sheaves with some additional structures. They lead the naturally defined semistability conditions, which are discussed in Section 3.3. We recall the concepts of Harder-Narasimhan filtration and partial Jordan-Hölder filtration in Subsection 3.3.2. Then, we introduce the notion of (δ,*l*)-semistability condition in Subsection 3.3.3, which is useful to control the transitions of moduli stacks of δ-semistable L-Bradlow pairs for variation of δ.

In Section 3.4, we review the boundedness of some families. In Subsection 3.4.1, we recall foundational theorems. Then, in Subsection 3.4.2, we recall the boundedness of δ-semistable L-Bradlow pairs when δ is varied. The important observation is due to M. Thaddeus. In Subsection 3.4.3, we show the boundedness of Yokogawa family, which will be used to show the properness of some morphisms in Chapter 4.

In Section 3.5, we recall 1-stability and 2-stability conditions. In Section 3.6, we recall some moduli schemes of quotient sheaves with some additional structures.

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