I am working on concrete examples of projective varieties over finite fields where one can verify Weil's conjectures. Following the ideas in Ireland-Rosen "A classical introduction to modern number theory" (Chapters 10-11) it is possible to compute the number of points of a hypersurface (living in $\mathbb{P}^n(k)$, where $k$ is a finite field with $q$ elements) defined by a homogeneous polynomial like $$ F(x_0,\dots,x_n) = a_1x_1^{c} + \dots + a_lx_l^{c} + a_{l-1}x_{l+1}^{c_l+1}x_0^{c-c_{l+1}} + \dots + a_{n}x_n^{c_n}x_0^{c-c_n} - ax_0^{c}, $$ where $a_i \neq 0$ and $a \in k$. In order words, $F$ is the homogenized of $$ a_1x_1^{c} + \dots + a_lx_l^{c} + a_{l-1}x_{l+1}^{c_l+1}+ \dots + a_{n}x_n^{c_n} - a. $$ However, in the book, only the case of Fermat hypersurfaces ($l=n$) is treated. My aim is to study which of this hypersurfaces satisfy Weil's conjectures and the first step is to find out which ones are smooth (which is one of the hypotheses of the conjectures). My computations lead me to the conclusion that
only the cases $l=n$ and $l=n-1$ are possible; and if $l=n-1$, then $c=2$ and $c_n=1$.
Does anyone know any reference where to find this or questions related to smoothness of similar hypersurfaces?