Algebraic geometry project ideas for high school students

Question

I am teaching a "senior seminar" course for strong students at our local high school. For 6 weeks the students learned about basic/classical algebraic geometry. In a few weeks they will start projects based on material from the course which they have to present. The idea is that the question they have to answer is difficult enough to be worth presenting, but not too difficult as to go beyond the scope of what was taught. Does anyone have algebraic geometry problems which would be good topics for projects?

I should mention what they have already learned in my lessons. We went through definitions and examples of affine and projective varieties (of course no mention of structure sheaves), regular and rational maps, projective space and it's standard affine covering, Bezout's theorem without proof, five points determine a conic, Segre and Veronese maps, singular and smooth points, blowing-up a point and basic resolution of singularities (for curves and some simple surfaces). I tried to stay clear of abstract/commutative algebra techniques since their knowledge in this area is limited.

One possible project I am thinking about is asking the student to explore/classify various types of singularities for curves and surfaces (cuspidal, nodal, pinch point etc) and learning some of the techniques for finding such singularities.

The five points determine a conic would have made a good portion of one project if I didn't already cover it in class.

Any ideas from past experience would be a big help.

UPDATE: The three project choices ended up being the following: An elementary proof that 27 lines lie on a smooth cubic surface, a basic exploration of resolution of singularities, and a proof of the Cayley-Bacharach Theorem. The students already presented a few weeks ago, and I was very pleased with the results. The material was definitely above what they were used to, but they learned a lot.

I thought it would be nice for future lecturers on the subject to know that high school students can learn the basics of the field.

Answer 1

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!

Answer 2

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:

1. 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.

2. 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.

3. In the spirit of 5 points determining a conic: the Cayley–Bacharach theorem, with details, and consequences, like Pascal's theorem.

Good luck!