To begin with, I am currently on what one could call elementary / classical algebraic geometry. OK in algebraic geometry almost everything is above elementary level ... but I don't imply simple. The level I mean is where one has a field $k$ and an algebraically closed field $K$ containing $k$ - and often the transcendence degree of $K$ over $k$ is infinite, then one says the first is an universal domain for the second. One then considers special subsets of affine space $K^n$ or projective space $(K^{n+1} \setminus \{0\}) / R$, where $R(x,y)$ is the equivalence relation (between $x$ and $y$) "there exists a non-zero $t\in K$ such that $y=tx$", any of these special subsets being the solution set of a system of polynomial equations with coefficients in $k$ (in the affine case, the projective case is a bit more complicated, but similar).
So please no answer mentioning schemes and higher. Best also no absolute varieties (let it be simple by staying "in $k$") nor abstract varieties (my question concerns affine & projective spaces which can be dealt without those, even if e.g. S. Lang introduces projective alg. geom. only in that frame - see below author+title for my two books on alg. geom.)
My question:
Introductory remarks. I appreciate the use of what is often called the Zariski ($k$-) topology (where the closed subsets are those algebraic over $k$) and the concept of a generic point of a $k$-variety can be defined from a general definition of a generic point in a topological space, using the Zariski topology. This works well in the affine case, i.e. leads to an equivalent algebraic property that agrees with the usual definition in algebraic geometry. For the projective case, I am less sure. This definition method works in both cases for the concept of irreducibility, it makes some proofs easier to expose and understand.
So I am facing this problem: For the definition of a generic point of a projective variety $H$ one can use:
a) the definition via the Zariski topology of projective space (which happens to be identical with the quotient topology of the topology induced from the Zariski topology of affine space to its part with 0 removed, as I could easily verify): in a topological space $X$ a point $a$ is by definition generic when the singleton $\{a\}$ is dense in $X$
b) the usual algebraic definition of a generic point as one finds it e.g. in the book referenced below as (2) specifically for projective space (after all, it cannot be the same as for affine space!) In the next § I try to "translate" the relevant passage of this book that could be the definition; from French but also from its mathematical presentation, which isn't quite clear to me any way (it is difficult to take it out of its context, but my translation might be a valid definition and that is then what I mean in what follows)
Let $\Pi$ denote the above mentioned n-dimensional projective space, $x \in \Pi$, and let $\hat x$ be a system of homogeneous coordinates of $x$ - which means coordinates of a point $P$ of the affine space above mentioned (with dim. $n+1$ instead of $n$) such that $x$ is the class of $P$ modulo $R$. We say that $x$ is a generic point of $H$ when we have:
A point $y \in \Pi$ belongs to $H$ if and only if it satisfies following statement: For every element $f$ of the ring of polynomials in $n+1$ indeterminates over $k$ such that $f(\hat x)=0$ and every system $\hat y$ of homogeneous coordinates of $y$, one has $f(\hat y)=0$. [I will later reference to this § under the name (Spec) for "specialisation" - it contains what Samuel names (Sp) but a bit more ...]
Comment to this definition: it seems to depend on the choice of $\hat x$ ... which makes little sense. BTW this passage is followed in the book by a statement that $\hat x$ is a system of strictly homogenous coordinates - a rather weird terminology which adds the restriction that $k(\hat x)$ is transcendental over $k(x)$ ... the author has first constructed (in the former context) a point $x$ and then says that such a point is said to be a generic point of $H$ - so this restriction might be a part of the definition (but probably not) - ... and finally he introduces a similar property using only homogeneous polynomials for f, and in this case the independance from the choice of $\hat x$ seems garantied ... really the whole is lacking clarity. I suppose others books say essentially the same but much clearer ...
Now: do a) and b) agree? If not, which definition is standard? If they do, could someone help me for the proof? One thing I know is that in one direction the implication works, due to a simple result I found: if a map from $X$ to $Y$ (top. spaces) is continuous and surjective, then the image of a generic point is a generic point. May-be a similar result could settle the question in the other direction.
As a justification to a), the German Wikipedia says in the article on this idea [my translation]: The concept of a generic point belongs to the mathematical subject of point-set topology, but it has its main application in algebraic geometry.
The tag "soft question" does not concern the whole question, it is here because definitions and the current standards about them can change. The part about possible equivalence of two definitions is not included in its scope.
References to books on algebraic geometry I am using:
(1) Serge Lang: Introduction to Algebraic Geometry
(2) Pierre Samuel: Méthodes d'algèbre abstraite en géométrie algébrique
The definitions are equivalent, of course.
For convenience, let me define some notation. Let $\mathbb{P}^n$ be projective $n$-space. For any subset $X \subseteq \mathbb{P}^n$, let $I (X)$ be the set of all homogeneous polynomials $f$ in $n + 1$ variables with coefficients in $k$ such that $f (x) = 0$ for all $x \in X$. Dually, for any set $F$ of homogeneous polynomials in $n + 1$ variables with coefficients in $k$, let $V (F)$ be the set of all $x \in \mathbb{P}^n$ such that $f (x) = 0$ for all $f \in F$.
(The condition $f (x) = 0$ does not depend on the specific homogeneous coordinates chosen for $x$: homogenous coordinates for a point in projective space are related by scalar multiplication, and homogeneity of $f$ means scalar multiplication of the input becomes scalar multiplication of the output, but scalar multiplication acts trivially on $0$. It is essential that we restrict attention to homogeneous polynomials, however.)
Recall that the Zariski topology on $\mathbb{P}^n$ is the one where the closed subsets are precisely those $X \subseteq \mathbb{P}^n$ such that $X = V (F)$ for some $F$. It is straightforward to verify the following:
Lemma. For $X \subseteq \mathbb{P}^n$, the closure of $X$ (with respect to the Zariski topology) is $V (I (X))$.
Thus, given a closed subset $H \subseteq \mathbb{P}^n$, $x \in H$ is generic in the sense of topology if and only if $H = V (I (\{ x \}))$. But $I (\{ x \})$ is by definition the set of homogeneous polynomials $f$ (etc.) such that $f (x) = 0$, so $V (I (\{ x \}))$ is the set of all $y \in \mathbb{P}^n$ such that $f (y) = 0$ for all homogeneous polynomials $f$ such that $f (x) = 0$. Hence $H = V (I (\{ x \}))$ is the statement that $x \in H$ is generic in the sense of algebra.
Let me remark that not every closed subset of $\mathbb{P}^n$ has a generic point. The ones that do are precisely the irreducible closed subsets.
(Let me also remark to scheme theorists: in this setup, generic points are not unique – indeed, there will be infinitely many of them! To get any generic points at all we need $K$ to be a transcendental extension of $k$, and to get generic points for everything we need algebraic closedness and infinite transcendence degree. This is different from the setup of e.g. Hartshorne Ch. I! It is essentially inspired by the practice of studying algebraic varieties defined over algebraic extensions of $\mathbb{Q}$ by looking at their $\mathbb{C}$-points – indeed, taking $k = \mathbb{Q}$ and $K = \mathbb{C}$ fits the requirements of this setup.)