Minimal Destabilizing Quotient

Question

For a pure $d$-dimensional sheaf $E$ on a projective algebraic variety over a field $k$, one has the Harder-Narasimhan filtration $$0\subset E_1\subset E_2\subset...\subset E_{l-1}\subset E_l:=E,$$ where the successive quotients $E_i/E_{i-1}$ are semistable with reduced Hilbert polynomials $p_i$ satisfying $p_{max}:=p_1>p_2>...>p_{l-1}>p_l:=p_{min}$. From the construction and uniqueness of the HN filtration, we know that $E_1$ is the maximal destabilizing sub sheaf of $E$, i.e. the unique subsheaf satisfying $p(E_1)\geq p(G)$ for all subsheaves $G\subset E$ such that $p(E_1)=p(G)$ implies $G\subset E_1$. The quotient $E/E_{l-1}$ is known as the minimal destabilizing quotient.

For some silly reason I'm having trouble showing that if $E\rightarrow G$ is any nontrivial quotient of $E$ then $p(G)\geq p_{min}$. Can anyone help me with this?

Also is there an intrinsic definition of the minimal destabilizing quotient?

Answer 1

This follows from the fact that $p(E)\geq p_{min}(E)$ for any pure sheaf. So if $E\rightarrow Q$ is a pure $d$-dimensional quotient, then we may compose with the surjection of $Q$ onto the last factor in its HN filtration, $G$. It follows from Lemma 1.3.3 in Huybrechts and Lehn's book on moduli of sheaves that we have $p_{min}(E)\leq p_{max}(G)=p_{min}(Q)\leq p(Q)$, as required.

I believe the required intrinsic definition would then be the unique pure $d$-dimensional quotient $Q$ such that any other pure $d$-dimensional quotient $G$ satisfies $p(G)\geq p(Q)$, where $G$ is a quotient of $Q$ in case of equality.