# Definition of Strongly Stable 0-cycle

I am not sure whether this question deserves to be asked in this forum, but I have no other choice as I can't find the definition anywhere. So here is the question:

When is a 0-cycle on $\mathbb P^n$ called strongly stable?

Mumford's "Lectures on Curves on an Algebraic Surface" contains a printing omission in P-141.

Here is what I wrote when the Stony Brook student seminar went through "Lectures on Curves on an Algebraic Surface".

Dear all,

At least in my edition, a critical definition was left off the bottom of p. 141 (probably a printer's mistake).

"An effective zero-cycle $A$ in $\mathbb{P}^n$ is 'strongly stable' if for every hyperplane $H$, the degree of $A\cap H$ is less than or equal to the total degree of $A$ divided by $(n+1)$."

If you participated in the GIT seminar a few semesters ago, this definition is related to, but stronger than, that definition of stability for zero-cycles, Def'n. 3.7, p. 73 of "Geometric Invariant Theory, 3rd ed." At first blush, it looks to me as though the analysis in Prop. 2 of Lecture 20 is very close to the analysis in Chapter 3, Section 2 of "GIT". I am not sure that helps understand the proofs (the arguments in both places are intricate), but perhaps it helps motivate the strategy.

As I mentioned in my last e-mail, I know of an alternative argument that avoids this analysis of 0-cycles, however it does have its own technical issues.

Best regards,

Jason