Some other interesting projective varieties — "flag varieties" — are also made of chunks called Schubert varieties, and the same reasoning applies to them. But most varieties are not so simple!
Elliptic curves illustrate the extra subtleties. As we've seen, they have both even- and odd-dimensional cohomology over \(\mathbb{C}\):
And over \(\mathbb{F}_q\) their number of points is not a polynomial in \(q = p^n\). It's
$$ p^n - \alpha^n - \overline{\alpha}^{\, n} + 1 $$
with \(|\alpha| = p^{1/2}\).
