Algebraic geometry project ideas for high school students
Maybe one of the possible projects would be about $27$ lines on a cubic surface. Usually the proof of the theorem uses some advanced algebraic geometry machinery, but there is a book by Reid which gives an elementary proof. I guess you could try to split it into a series of exercises. Maybe this is far from doing reseach kind of project, but it might work, and I think the result is mesmerizing. The book itself is actually quite nice, and it has a lot of exercises. Maybe you can find something else there that can be turned into a high school project.
I hope this helps a bit!
First of all, what lucky students! I wish I had been offered such a course when I was a high-school student.
Second, I must admit I don't have previous experience of such a project, so you should take my recommendations with a pinch of salt.
Here are some ideas that occurred to me:
Grassmannians, including an epsilon of Schubert calculus. For example, you could get them to prove that there are exactly two lines touching 4 general lines in $\mathbf P^3$, with and without Schubert calculus.
Elliptic curves. Of course, there are many directions you could go in: for example, proving that the chord-tangent construction really defines a group structure; alternatively, if they know a bit of complex analysis, the Weierstrass function, and proving that an elliptic curve is topologically a torus.
In the spirit of 5 points determining a conic: the Cayley–Bacharach theorem, with details, and consequences, like Pascal's theorem.
Good luck!