I am currently studying this paper, and I am not being able to prove one argument on the proof of the Theorem $5.1$, namely, the last part where I have to show that a product of divisors is bigger than zero.
On the proof we have $\alpha$ a curve on the Hilbert scheme $\mathbb P^{2[n]}$, constructed as each $Z\in \alpha$ being a complete intersection of polynomials of degrees $a,b$, and $D_E=\{Z\in\mathbb P^{2[n]}|h^0(I_Z\otimes E)\neq 0 \}$, where $E$ is a fixed vector bundle. I know that $D_E= \mu(E)H-\frac{1}{2}B$, where $H=h^*\mathcal O_{\mathbb P^{2(n)}}(1)$, $h$ the Hilbert-Chow morphism, and $B$ the exceptional divisor of $h$ parametrizing non-reduced schemes. Since I know that $\alpha D_E=0$, I want to show that the divisor associated to a bundle $E'$ with smaller slope $\mu'$ satisfies $\alpha D_{E'}<0$, to do this I wish to show that $\alpha H>0$, so this would solve the last part of the theorem. I was suggested to try to use the projection formula for cycles $g_*(f^*D\alpha)=D f_*\alpha$, but I am not managing to solve the problem. Any hints?
Thanks in advance.
Your statement of the projection formula doesn't seem right: it has both an $f$ and a $g$ in it, but the projection formula only involves one morphism.
If we write it correctly, and with $\cdot$ denoting intersections, it should look like
$$ f_* ( (f^*D) \cdot \alpha) = D \cdot f_* \alpha.$$
Now the only morphism you have in your setup is $h$, so probably you should apply the formula with $f=h$. Next, what is $D$? Well, you want to calculate $H \cdot \alpha$, and by definition $H = h^* O(1)$ so you should take $D$ to be $O(1)$.
So now you need to show that $O(1) \cdot h_* (\alpha) >0$. How to do that?
Well, by definition $O(1)$ is an ample line bundle on $(\mathbf P^2)^{(n)}$, the $n$-th symmetric product of the plane. An ample line bundle has positive intersection with every curve. So it is enough to prove that $h_*(\alpha)$ is a curve --- in other words, that $\alpha$ is not contracted to a point by $h$. But when is a subset of the Hilbert scheme contracted by $h$?
I will leave this last step to you.