Intersection number of divisors with its pull back and its push forward

Question

I am in an ideal situation but I would appreciate a hint. First here is the scenario.

Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ which has one $k$-rational point $\infty$ . Consider $\Theta\subset\mathcal{A}$ given by the the image of the Abel-Jacobi map $\iota:\mathcal{H}\to\mathcal{J}$ where $\iota(P)=[P-\infty]$. We have that $\Theta$ can be regarded as a divisor in $\text{Div}(\mathcal{J})$ which is ample since a basis of $\mathcal{L}(4\Theta)$ defines an embeding of $\mathcal{J}\hookrightarrow\mathbb{P}^{15}$.

The question:

Let $D_1,D_2\in\text{Div}(\mathcal{J})$. We denote by $D_1\bullet D_2$ the intersection number of the divisors $D_1$ and $D_2$ (In this case working with the abelian variety $\mathcal{J}$ of dimension $2$, the divisors $D_1$ and $D_2$ can be regarded as algebraic curves inside $\mathcal{J}$, therefore $\bullet$ denotes the cardinality of $D_1\cap D_2$ as curves inside $\mathcal{J}$ with counted multiplicities).

Let $\gamma\in\text{End}_k(\mathcal{J})$ I want to know if under all these hypotheses the following two integers are equal:

$ \gamma^*\Theta\bullet\Theta=^?\gamma_*\Theta\bullet\Theta $.

And if not, how they are related.

Here $\gamma^*$ and $\gamma_*$ are the pull back and push forward maps from $\text{Div}(\mathcal{J})\to\text{Div}(\mathcal{J})$.

I have checked in Fulton's but I cannot find some example or theorem what helps me deducing this but this should be something already known.

Answer 1

There is a very general projection formula that's valid for any proper morphism $f:X\to Y$ and any cycles $x$ and $y$ in the Chow groups of $X$ and $Y$ respectively: $$ f_*(x\cdot f^*y) = f_*(x)\cdot y. $$ In your case, the intersections are 0-cycles, so taking the degree of both sides gives the formula that you want. You can find the projection formula stated in Hartshorne Algebraic Geometry, Appendix A, Property A4, for example.