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Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$. By the Lang-Weil bounds $ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ depending on $n$, $d$ and $N$.

Are there any bounds on $C$? Can you improve the estimate in special situations? I am interested in the case of del Pezzo surfaces where I want to use to test for local solubility.

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Del Pezzo surfaces are rational and you can work out their zeta function explicitly. This should be in Manin's book on cubic surfaces. – Felipe Voloch Aug 18 '13 at 12:48
In general, one gets much better bounds for smooth projective vareties using Deligne's proof of the Weil conjecture. For a del Pezzo surface $X$ you get that the error term is bounded by $bq + 1$ where $b$ is the second Betti number of $X$, which is the number of points of $\mathbb{P}^2$ blown up to get $X$ (over an algebraic closure of $\mathbb{F}_q)$ plus $1$ (so between $1$ and $9$). – ulrich Aug 18 '13 at 12:56
Dear Ulrich, is there any reference for that result (besides reading the proof ...)? Thanks! – Casaubon Aug 18 '13 at 13:43
Like I said, Manin's book. Theorem 27.1 and corollary 27.1.1. – Felipe Voloch Aug 18 '13 at 15:02

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