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.