# What is meant by a set of generic points on a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus $g \geq 1$.

I'm studying a theorem of Faltings which looks as follows.

Let $P_1,\ldots, P_g$ be generic points on $X$. Then we have some equality concerning theta functions. (Details given below in Edit.)

What does it mean that $P_1,\ldots,P_g$ are generic points?

It means that the points don't lie on the theta divisor.

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Perhaps they just mean points in general position. A compact connected Riemann surface should have only one generic point (in the sense of algebraic geometry). –  Zhen Lin Oct 1 '11 at 14:35
What do you mean by points in general position? It is true that the algebraic curve associated to a compact connected Riemann surface has precisely one generic point (simply because it's irreducible). –  shaye Oct 1 '11 at 14:46
"in general position" means that "such that all the conditions needed to do what I am about to do hold" and is only used when the set of all choices is big (for example, contains a dense open set) –  Mariano Suárez-Alvarez Oct 1 '11 at 17:36
@MarianoSuárez-Alvarez. Ok so that's not what I'm looking for. –  shaye Oct 1 '11 at 18:10
@shaye, my guess is that that is what you are looking for, but it is impossible to tell given the lack of details. –  Mariano Suárez-Alvarez Oct 1 '11 at 21:16