I'm actually seeking a formal definition of a general point. I already read the post in https://mathoverflow.net/questions/64089/general-point-general-line/64096 but it was not helpful. In addition, several Algebraic geometry references uses this as a well-known notion, but I don't know what is the formal formulation for it.
Anyone can help me with a reference or a definition? Thanks in advance.
K.Y.
I'm sure there are tons of different people over the web and in books, lectures, etc. explaining this simple but elusive concept. I'll also give a shot, trying to assume as little about your algebraic geometry knowledge as possible.
First of all, there is no such thing as a standalone "general point". A general point is always accompanied with a property. The idea of a general point is not even special to algebraic geometry. Here is the idea with a down to earth non-geometric example:
Consider a box containing 1 gazillion balls. There are 10 red balls in there and the rest are blue. If you take a random ball out of the box, chances are its color is blue. One says: our box generally contains blue balls. This does not mean there is no red balls in there, only that almost every ball is blue.
In mathematics, the concept almost every is carefully defined depending on the context and is only meaningful for topological spaces. A set $S$ in a topological space $X$ is called dense if given any point $x\in X$ and any open neighborhood $V$ of $x$, one has $V\cap S\neq \emptyset$. In other words, any point in $X$, if not in $S$ already, is arbitrarily close to it (there is no escaping $S$, it'll be in your face no matter what).
Now let us collect all points in $X$ having some property $P$ inside a set $S$. For example, consider the curve $f(x,y)=y^2-x^3=0$. A point is called singular if $\partial f/\partial x=\partial f/\partial y=0$. Clearly, the only singular point of this curve is at the origin. One then says A general point of $y^2=x^3$ is non-singular, or less conventionally, but maybe more intuitively: The points of $y^2=x^3$ are generally nonsingular.
Now in algebraic geometry one already has a topology, Zariski topology. Open sets of Zariski topology are quite "bulky". For example, given an affine variety (over an algebraically closed field), all Zariski open sets are dense. So this is what people usually do:
Say you are interested in some property $P$ of the points of a variety/scheme $X$ (like nonsingularity above). One first gathers all points $X$ satisfying $P$ in a set $S$. If $S$ is a
Then we say a general point of $X$ satisfies property $P$. In other contexts that I discussed, we did not demand $S$ to be an open set. But it turns out it is much more convenient and meaningful to also implement this "openness" condition in algebraic geometry.