TITLE: Effective Lang-Weil bounds for del Pezzo surfaces
QUESTION [5 upvotes]: 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.

REPLY [7 votes]: I hope nobody minds me answering this (very old) question.
Me and my collaborators (Barinder Banwait and Francesc Fité) succeeded in completely answering this question in the paper:
Del Pezzo surfaces over finite fields and their Frobenius traces (https://arxiv.org/abs/1606.00300).
Specifically, let $S$ be a del Pezzo surface of degree $d$ over a finite field of size $q$. Then
$$\#S(\mathbb{F}_q) = q^2 + aq + 1$$
for some $a \in \mathbb{Z}$. Then in our paper we completely classified which values of $a$ can occur for fixed $d$ and $q$.
For example for cubic surfaces we have $a \in \{-2,-1,0,1,2,3,4,5,7\}$, and each value occurs over every finite field, except for $a = 7$ which does not occur when $q = 2,3,5$.
Other degrees $d$ are similar, but the list of admissible values of $a$ and the number of exceptional cases in small finite fields is quite long.