Let $\mathbb F_q$ be a finite field with $q$ elements where $q$ is a power of a prime $p$. Consider the polynomial ring

$$\mathbb F[x_{ij}],$$ for $i,j=1,\ldots ,n$. Let $f$ and $g$ be polynomils in $\mathbb F[x_{ij}]$ defined by $$f=\sum_{\sigma\in S_n}sgn(\sigma)x_{1\sigma(1)}\ldots x_{n\sigma(n)}$$ and $$g=\sum_{\sigma\in S_n}x_{1\sigma(1)}\ldots x_{n\sigma(n)},$$ where $S_n$ denotes the set of all permutations of the set {1, 2, . . . , n} and $sgn$ is the signum of the permutation $\sigma$.

Let $X$ and $Y$ be the projective homogeneous varieties defined by $f$ and $g$, respectively.

how can I prove (I feel that it's true) that the varieties $X$ and $Y$ have not the same number of points?