2
$\begingroup$

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.

  • 0
    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).2011-10-01
  • 0
    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).2011-10-01
  • 2
    "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)2011-10-01
  • 0
    @MarianoSuárez-Alvarez. Ok so that's not what I'm looking for.2011-10-01
  • 0
    @shaye, my guess is that that *is* what you are looking for, but it is impossible to tell given the lack of details.2011-10-01
  • 0
    I added some details. I don't think this is what I'm looking for because what would "all the conditions needed to do what I am about to do" mean in this context???2011-10-02
  • 0
    @shaye: I'm just guessing here, since I'm not familiar with this material, but could it be that one of the terms in the equation is not defined for some special configurations of the points?2011-10-02

0 Answers 0