If you like characteristic p, Pardini's characterization of smooth plane curves all of whose points are inflections (Compositio 60 (1986) 3-17) is very nice and uses only Bezout's theorem, basically. Bonus #1: it was a tesi di laurea (Italian equivalent of a senior thesis). Bonus #2: author is on MO http://mathoverflow.net/users/10610/rita

If you like characteristic p, Pardini's characterization of smooth plane curves all of whose points are inflections (Compositio 60 (1986) 3-17) is very nice and uses only Bezout's theorem, basically. Bonus #1: it was a tesi di laurea (Italian equivalent of a senior thesis). Bonus #2: author is on MO https://mathoverflow.net/users/10610/rita