# Density of rational points over finite fields

Let $$X\hookrightarrow\mathbb P^n_{\mathbb F_q}$$ be a geometrically integral hypersurface over the finite field $$\mathbb F_q$$ of degree $$\delta$$. In order to estimate the number of its rational points, we have $$|\#X(\mathbb F_q)-(q^{n-1}+\cdots+1)|\leq(\delta-1)(\delta-2)q^{n-\frac{3}{2}}+C(n,\delta)q^{n-2}.$$ My question is about the order of $$\delta$$ in the constant $$C(n,\delta)$$. In fact, we have $$C(n,\delta)\ll_n\delta^4$$ uniformly, and when $$q$$ is large enough, we can prove $$C(n,\delta)\ll_n\delta^2$$. Can we prove $$C(n,\delta)\ll_n\delta^2$$ uniformly for all $$q$$?

If it is too difficult, we can consider whether we can prove $$|\#X(\mathbb F_q)-(q^{n-1}+\cdots+1)|\leq C'(n,\delta)q^{n-\frac{3}{2}},$$ where we require $$C'(n,\delta)\ll_n\delta^2$$ uniformly. Of course, this is OK for the case of $$n=2$$, but for arbitrary dimension, I don't know......