The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$. Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be the projective hypersurface defined by $f = x_0^d + ... + x_n^d$. Let $\mathcal{M}_r = \mathcal{M}_r(\mathbb{P}^1_k, \mathfrak{X})$ be the space of degree $r$ rational curves on $\mathfrak{X}$. We define $$ P_r = \{ x \in k[u,v]: \text{homogeneous and of degree } r \}. $$ If $V$ is a scheme of finite type over a finite field $k$ and $R$ is a finite $k$-algebra we denote $$ [V]_R = \#V(R)/ \#R^{\text{dim }_e V}, $$ where $\text{dim }_e V$ is the "expected dimension" of $V$ (It is not well defined in the thesis either). But to give an example, for $$ M_r = \{ (x_0, ..., x_n)\in P^{n+1}_r : f(x_0, ..., x_n) = 0\}, $$ the expected dimension is given by $(r+1)(n+1)-(rd+1)$.
I have two questions: (1) What would be the expected dimension of $[\mathcal{M}_r]$?
(2) On page 28, he shows that $$ [M_r] = (1- q^{-1})\sum_{\deg D \leq r} [\mathcal{M}_{r - \deg D}], $$ where $D$ is an effective divisor. From here, he says "Expanding into an Euler product, we get $$ [M_r] = (1- q^{-1}) [\mathcal{M}_{r}] \zeta_{\mathbb{P}^1}(q^{d-n}) + O(q^{r(d - (n+1))})," $$ for which I am stuck on. Could anyone possibly give me a hint or explanation so that I can figure this out?
I would greatly appreciate any help understanding this. Thank you very much!
