Here follows a list of possible topics for further study that can replace a written or oral exam. The topics are of different degree of difficulty. Anyway, the degree of difficulty also depends heavily on how far one is willing and able to go. Consider it just as a list of suggestions. The best is to contact me after you have made your choice, and also when you cannot make your choice.

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1) Weierstrass points on Riemann surfaces Literature: Forster:p 150-153. Can you do the Exercises of [A-C-G-H], p. 41-44 on Weierstrass points ?
2) Do the exercises of [A-C-G-H] on Automorphisms of Riemann Surfaces, p 44-47.
3) Read Chapter I of [A-C-G-H], p 1-30.
4) The Jacobian of a Riemann surface. Literature: Forster: Sections 20 & 21. For the Theta divisor, see: [G-H], p. 333-340 and beyond. Alternatively: Gunning, Section 8, (p. 129-163).
5) The analogy between Riemann surfaces and algebraic number fields. Literature: [N] Ch. III (Riemann-Roch theory), or [vdG-S].
6) Analyze canonical curves of genus 4 in projective threespace. Do this yourself; if you don't succeed see [A-C-G-H].
7) Modular curves as Riemann surfaces, see [D-S], Ch. 2-3; [Gu].
8) Riemann surfaces of genus 2 and Kummer surfaces. See the corresponding chapters of [G-H].
9) Riemann surfaces of genus 3, bitangents and cubic surfaces

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[A-C-G-H] E. Arbarello, M. Cornalba, Ph. Griffiths, J. Harris: Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften, 267. Springer-Verlag, New York, 1985
[D-S] F. Diamond, J. Shurman: A first course in Modular forms. Springer Verlag
[Gu] R. Gunning: Lectures on Riemann surfaces.
[G-H] Griffiths, Harris: Principles of Algebraic Geometry.
[N] J. Neukirch: Algebraische Zahlentheorie. Springer Verlag, 1992. An English translation exists too.
[vdG-S] : G. van der Geer, R. Schoof: Effectivity of Arakelov divisors and the theta divisor of a number field. Selecta Math. (N.S.) 6 (2000), no. 4, 377--398. (math.AG/9802121).