Let $X$ be a smooth connected scheme over field $k$. It is known that
correspondences $\alpha \subset X \times X$ regarded as 
objects in Chow groups $\text{CH}^*(X \times X)$ 
act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", 
namely we have a cycle class map 

$$ \text{cl}:  \text{CH}^*(X) \to H^*(X, \mathbb{Z} )  $$

which is compatible with pull-backs & push-forwards. If
$p_1, p_2: X \times X \to X $ are the projections and $\alpha $
a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ 
\alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by *rational* 
Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q}  $
on rational cohomology.


Let now specialize: Assume $X$ is a curve and $D \subset X$ 
an effective divisor of degree $d$, ie a "multisection".
By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a 
$\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the [discussion here][1]
Dan Petersen uses that the induced action by 
such $\mathbb{Q}$-divisor of degree $1$ in terms from above 
as associated Chow correspondence 
induces a splitting of the cohomology.


Question: I not see why this action gives a splitting of the cohomology?
In other words does this $\mathbb{Q}$-divisor
induce an idempotent endomorphism corresponding 
to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection.  
(Dan Petersen used this more generally in relative setting
& associated action on derived object $Rf_* \mathbb{Q}$ but for 
sake of simplicity I would like to understand it in absolute case for a 
single curve.)


  [1]: https://mathoverflow.net/q/443274