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I'm currently reading Beauville: complex algebraic surfaces as a start for my master thesis in algebraic geometry. Before, i did two courses on AG, basicaly chapters 1 and 2 of Hartshorne and some cohomology theory.

Now Beauville uses the term "generic" every now and then, and i'm having a hard time finding out if he means a property of a generic point of a variety, or a property that one expects to hold for a random point. So a first question could be: how is this term used normally in algebraic geometry? However i have more specific questions:

By a generic point i will mean a non closed point on n dimensional projective space (so a non maximal prime ideal). Pick a point p in $\mathbb{P}^n$, and consider projection away from p. Question: What happens if p is not a closed point? My guess is that a line L trough p must contain $\overline{p}$, which we for the moment suppose to be a line, hence $L = \overline{p}$? So there is just one line through p?? But then projection from p does not make sense. And if the dimension of $\overline{p}$ is bigger then one, then a line through p has dimension bigger then one? That does not make sense at all. However, Beauville page 42, prop IV.5, talks about "generic projection". Question: So does he mean projection away from the generic point, or does he mean generic in the sense of "almost all points" here? If my reasoning above is correct, it will of course be the last, but it very well might be flawed.

So far for generic. My second and last question concerns degree of varieties. On the same page, p.42, Beauville talks about a surface S in $\mathbb{P}^n$. He lets H be a hyperplane section of S, and then says H.H = deg(S) (intersection product of curves) Question 1: I assumed that by a "hyperplane section of S", he means the intersection of S with a hyperplane (assumed to be nonempty). Is this correct? Question 2: Is it true in general that the intersection product of such a hyperplane section with itself gives the degree of a surface?

The questions seem really specific, but answers to them will help me in a lot of other places in the book as well! So lots of thanks in advance!! Joachim

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It seems from your description that he uses generic to mean both a generic point and point in general position. Other than projections onto factors from fibre products, the only projections I've encountered are away from closed points. So I think he must be talking about projections from points in general position. – Andrew Jul 18 '12 at 17:39
I.e. a generic member of the family of all projections from (closed) points in $\mathbb P^n$. – Andrew Jul 18 '12 at 17:50
Dear Joachim, I don't see anything about hyperplane sections on page 42. I checked the second edition of the book: what is your edition? Meanwhile I have given an answer to the first question below. – Georges Elencwajg Jul 18 '12 at 20:26
Dear Georges, thank you so much for your help. I bought the book just a few months ago, it's the second edition as well. Maybe this helps: its Fact IV.4 (4) (a), second sentence. – Joachim Jul 19 '12 at 9:20
Edit: If its not there in your copy, i can summarize it as follows. Take a rational map $P^2$ -> $P^N$ (derived from some linear system on P^2), blow up $P^2$ to a surface S s.t. we derive a morphism $f: S -> P^N$, assume it is an embedding and put S' = f(S). Let H be a hyperplane section of S', then we have H.H = deg(S'). The last statement i dont understand so that is my question. It may have to do with the linear system P on P^2 consisting of divisors of some fixed degree d, so that $H = dL - \sum m_i E_i$ where the $E_i$ are the exceptional divisors and $L$ comes from a line on $P^2$. – Joachim Jul 19 '12 at 9:36
up vote 3 down vote accepted

Beauville in his book uses "generic" for a property holding for (at least) all points in a non-empty open subset of the variety under study.
Such an open subset is the complement of a variety and the proof of Lemma IV.4. describes such varieties in $\mathbb P^N$ explicitly.
Generic projection thus means projection from any point except those on some closed subvariety of $\mathbb P^N$.

The book is written in the pre-scheme, classical, language of varieties and doesn't use non-closed points, so don't worry about projecting from them!

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Wow that settles one of my questions completely, thanks a lot! – Joachim Jul 18 '12 at 20:25
But it raises another one though, somewhere earlier in the book Beauville talks about a map being "generically finite of degree d". Hartshorne's definition of this is that the fibre of the generic (non closed) point consists of d points, but i understood that the classical definition was that it induces a map of function fields that is an extension of degree d (which implies (i dont know why) that the set of points that have d points (with multiplicity) in the fiber is nonempty open). Could you confirm that these two definitions are equivalent? – Joachim Jul 18 '12 at 20:33
Dear Joachim, what you write is convincing but I can't guarantee it. By the way, De Jong, one of the great contemporary algebraic gemeters writes that he doesn't know what the correct definition of "generically finite " should be in general:see the comments here – Georges Elencwajg Jul 18 '12 at 21:03

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