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I am trying to understand an example in the paper Random hypersurfaces and embedding curves in surfaces over finite fields by Joseph Gunther.

For a scheme $X$ of finite type over a finite field $\mathbb{F}_q$, let $\zeta_{X}^{[l]}(s) := Z_X^{[l]}(q^{-s})$, where $Z_X^{[l]}(s)$ is defined in this question.

For $X= \mathbb{P}_{\mathbb{F}_q}^2$, we have $\zeta_{X}^{[1]}(s) = \frac{q^2+q+1}{q^s -1}$.

I don't understand how this was calculated. Also, how do I go about computing $\zeta_{X}^{[l]}(s)$ in general for $X=\mathbb{P}^n_{\mathbb{F_q}}$? I sort of understand the definition but I need to see an explicit computation to make things clear.

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    Hint: each $\lambda_i$ is at least one, so if $\sum_{i=1}^r \lambda_i=1$, that means $r=1$ and $\lambda_1=1$, so ${P_1}$ is just a rational point. How many rational points are there on $\Bbb P^2$? Do you see how to apply the result from your previous question now? – KReiser Apr 20 '20 at 00:17
  • @KReiser I see. There are $q^2 + q + 1$ rational points in $\mathbb{P}^2$ and each of them contributes $\frac{1}{q^{s}-1}$ to the zeta function. Now we just have to add all of them up. – I like Cake Apr 20 '20 at 09:08
  • Yes, that's exactly it. I suggest recording this as an answer to your question below. – KReiser Apr 20 '20 at 09:21

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As explained in the comments, we need to consider $r$ distinct closed points $P_i$ of degrees $\lambda_i$ such that $\sum_{i=1}^r \lambda_i = 1$. This implies $r = \lambda_i = 1$. So we have a single closed point which is rational. And the contribution of zero-cycles supported on this rational point to the zeta function is $\frac{1}{q^{s} -1}$. Since the number of rational points is $q^2 + q + 1$, we get the result.

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