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Are there absolute constants $N$ and $B$ such that the following is true?

Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with coefficients in $\mathbb{F}_p$, where $n>N$. Let $V \subset \mathbb{P}^n$ be the variety given by $$ V \; : \; q(x_0,\dotsc,x_n)=c(x_0,\dotsc,x_n)=0. $$ Here I don't assume that $V$ is irreducible, non-singular, non-degenerate, etc. Suppose that $V$ is not contained in the hyperplane $x_n=0$. Show that there is a point $(a_0:\dotsc:a_n) \in V(\mathbb{F_p})$ so that $a_n \ne 0$. Equivalently if we identify $\mathbb{A}^n$ with $x_n=1$, then we would like to show the existence of a $\mathbb{F}_p$-point on the affine variety $V \cap \mathbb{A}^n$.

For a fixed $n$, it is possible to use Lang--Weil estimates to come up with a suitable $B$. However, the value of $B$ will depend on $n$.

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    $\begingroup$ Won't you have trouble if $q=x_0^2-dx_n^2$ and $d$ is a non-square regardless of $n,N,B$ or $c$? $\endgroup$
    – Felipe Voloch
    Commented Aug 16, 2014 at 11:57
  • $\begingroup$ You're right Felipe! I have to rethink what I want. $\endgroup$
    – Siksek
    Commented Aug 16, 2014 at 13:48
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    $\begingroup$ What you want may relate to Chevalley's theorem: en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem $\endgroup$
    – Will Sawin
    Commented Aug 16, 2014 at 14:34
  • $\begingroup$ Thanks Will. Chevalley's theorem shows that the number of affine solutions is divisible by $p$. I don't see how to use this to deduce that the number of affine solutions is positive! $\endgroup$
    – Siksek
    Commented Aug 16, 2014 at 14:55

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