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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.