Recent, elementary results in algebraic geometry

Question

Next semester I will be teaching an introductory algebraic geometry class for a smallish group of undergrads. In the last couple weeks, I hope that each student will give a one-hour presentation. The usual approach here might be to suggest some nice classical stories (e.g. stuff in this thread) and have each student pick one.

I'm hoping to convince the class that algebraic geometry, even as it is currently practised, is not such a frightening field as they have been led to believe -- that there is still some low-hanging, elementary fruit. To that end, I'd like to mix in some more recent results, say papers appearing in the last year or two. I'm looking for possible topics.

Question: Can anyone suggest recent papers s.t. (1) the statements are appealing to novices (2) the statements can be understood by an undergrad familiar with the material in Shafarevich's book (no schemes, no derived categories, no toric varieties, no moduli spaces... you get the idea) (3) the outline of the proofs could conceivably be understood and presented by said undergrads.

These do not necessarily need to be ground-breaking papers in big deal journals. Minor results are fine, as long as they plausibly sound interesting to non-experts. A quick skim suggests the median number of suitable papers per day on math.AG is 0.

Any suggestions? Self-promotion is welcomed.

Answer 1

This paper showed two century-old classification results, each of very undergrad-comprehensible things, were the same; pretty amazing!

http://arxiv.org/abs/1308.0751

"Sums of squares and varieties of minimal degree" by Grigoriy Blekherman, Greg Smith, and Mauricio Velasco

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree.

Answer 2

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

Answer 3

I know you said no moduli spaces, but this paper would be great for a reading project. The moduli spaces in question is not complicated to construct, and the proofs are accessible with guidance.

http://arxiv.org/abs/0901.1783

Cheers!

Answer 4

You don't get more algebraic geometry than understanding the equations defining a (affine or projective) variety, and there are still tons of questions being actively studied. For example:

1) Google "set-theoretic complete intersections". While you are at it, also search for "monomial curves".

2) "Secant varieties". These even have connections to statistic, etc, search for "tensor decomposition" and all that. Even dimensions of these objects are non-trivial problem, as far as I understand.

3) Commutative algebra usually has quite a few accessible papers, and many problems there have strong links to algebraic geometry. So visit math.AC. Good luck!