I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold theory, but may not have had much expose to algebraic geometry. I will loosely follow the book  [Introduction to Algebraic Curves](http://rads.stackoverflow.com/amzn/click/0821845373) by Griffiths. In particular, I hope to spend a minimum amount of time developing basic machinery (e.g. sheaf theory) and to start doing concrete geometry (e.g. canonical models of curves of genus up to 4) as soon as possible. 


My question is: what are some good concrete, accessible geometric topics in Riemann Surface/Curve theory that aren't in the standard textbooks? 


Let's say that the standard textbooks are the book I mentioned and those discussed:
  [here.](https://mathoverflow.net/questions/1916/riemann-surfaces) 



What should be taught in a 1st course on Riemann Surfaces?
Good question. I bet you'll get many interesting answers.


About two years ago I taught an "arithmetically inclined" version of the standard course on algebraic curves. I had intended to talk about degenerating families of curves, arithmetic surfaces, semistable reduction and such things, but I ended up spending more time on (and enjoying) some very classical things about the geometry of curves. My lecture notes for that part of the course are available [here](http://alpha.math.uga.edu/%7Epete/8320notes6.pdf):


Some things that I found fun:


1. Construction of curves with large gonality. For instance, after having given several examples of various curves, it occurred to me that I hadn't shown them a non-hyperelliptic curve in every genus g >= 3, so then I talked about trigonal curves, and then...Anyway, there is a very nice theorem here due to Accola and Namba: suppose a curve $C$ admits maps $x,y$ to $\mathbb{P}^1$ of degrees $d\_1$ and $d\_2$. If these maps are independent in the sense that $x$ and $y$ generate the function field of the curve (note that this must occur for easy algebraic reasons when $d\_1$ and $d\_2$ are coprime), then the genus of $C$ is at most $(d\_1-1)(d\_2-1)$.


I sketched the proof in an exercise, which was indeed solved in a problem session by one of the students.


2. Material on automorphism groups of curves: the Hurwitz bound, automorphisms of hyperelliptic curves, construction of curves with interesting automorphism group.
3. Weierstrass points, with applications to 2) above.


