Segre's theorem

Remarks

The theorem was proved by Beniamino Segre. The key step is the "lemma of tangents" for an oval in $\mathrm{PG}(2, q)$ with $q$ odd. It says that any triangle of points of the oval and the triangle of tangent lines at these points are in perspective. Its proof uses that the product of all nonzero elements of $\F_q$ is $-1$. From this one shows that the oval agrees with the conic through five of its points.

The hypothesis that $q$ is odd is necessary. Let $q = 2^h$ with $h \ge 3$, let $C$ be a conic and let $N$ be its nucleus, the common point of all its tangent lines. Removing a point $P \ne N$ of $C$ from $C \cup \set{N}$ gives an oval. It is not a conic: it shares $q \ge 8$ points with $C$, while two distinct conics share at most $4$ points.

Going deeper

For a student

Segre's theorem is the starting point of Galois geometry, the study of point sets in projective spaces over finite fields. It shows that in $\mathrm{PG}(2, q)$ with $q$ odd the ovals have a single type up to projective equivalence. In even characteristic the picture is different: every oval extends to a hyperoval of $q + 2$ points, and many non-conic examples exist. The same circle of ideas classifies ovoids of $\mathrm{PG}(3, q)$ for $q$ odd as elliptic quadrics.