All of the sources that I have found characterizing very ample divisors reference Hartshorne or otherwise use, in my opinion, “non-elementary” arguments. Here are some “elementary” proofs that I worked out with some fellow grad students.
Belyi’s Theorem is an arithmetic-geometric result that has the unusual quality of being both fairly recent and provable with only the material from a first course in classical algebraic geometry.
I have spent a lot of time applying to grad schools, here are some useful things that I found during my search.
A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In my computer science masters project I implemented several algorithms to compute path colorings of graphs embedded in the plane.
The domain name “permutation lock” is something that I came up with back in high school when I learned that the locking mechanism on a “combination lock” does not fit with the combinatorial notion of a combination.