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I have learned that in algebraic geometry, when an 'object' can be put in a family which is in a bijective correspondence with some projective variety, the generic object in this family is one which lies in a (Zariski) open dense subset of the variety.

So, for example we can talk of the generic divisor in a linear system, e.g. the generic curve on a surface etc.

However, for curves I have seen the word general being used as well, sometimes. I initially thought that this were synonyms, but then started wondering if this is really so. I am a bit confused.

I did not find an answer by looking at the index in Hartshorne's book. Can somebody illuminate me on this terminology? What is meant by a general curve, or a general curve of genus $g$ (e.g. in the first page on this article) or general in the sense of moduli? are there other uses of general?

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2 Answers 2

up vote 6 down vote accepted

They are not exactly synonyms.

A generic property is a property of the generic point. A general property is a property that holds away from a Zariski closed subset. To illustrate the difference, I would make the following:

Example. If you have a linear system of divisors $|D|$, you can say that the generic element of $|D|$ satisfies a certain property $Q$ if the generic point of $|D|$ (which is a projective space, hence has a unique generic point) satisfies $Q$. On the other hand, you can say that the general member of $|D|$ satisfies $Q$ in case there is a Zariski dense open subset $U\subset |D|$ such that every point in $U$ satisfies $Q$.

Remark. If $F$ is a scheme parametrizing a certain family of schemes, and $Q$ is a property of schemes, let us consider the sentences:

  1. The generic element of $F$ satisfies $Q$;
  2. The general element of $F$ satisfies $Q$.

The negations of 1 and 2 are sensibly different:

  1. The generic element of $F$ does not satisfy $Q$;
  2. The subset of $\{p\in F\,|\,p\textrm{ satisfy }Q\}\subset F$ has empty interior.

To sum up: if a property holds for a general element, it does not mean it holds for the generic point (provided that you have one such). But sometimes one can, in some sense, go the other way round:

From generic to general. Suppose you have a morphism of schemes $X\to Y$, with $Y$ irreducible of generic point $\eta$. Let $Q$ be a property of schemes. If the generic fiber $X_\eta$ has $Q$, and the property is constructible, then a general fiber has $Q$ as well, meaning that there is an open subset $U\subset Y$ such that $X_u$ has $Q$ for every $u\in U$.

Curves. I guess by "a general curve" of genus $g$ one means a general point in $M_g$. Of course $M_g$ is an irreducible variety, so it also has a unique generic point, and it makes sense to make statements about the generic curve of genus $g$.

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I think this has been answered before on the site. Anyway, here is the summary:

General means lying in some Zariski-dense open subset of the parameter space of objects in question. In other words, "property $P$ holds for a general object" means that there is a Zariski-dense open subset $U(P)$ (depending on $P$, of course!) such that $P$ is true for objects corresponding to points in the parameter space which lie in the subset $U(P)$.

Very general means lying in the complement of the union of countably many proper closed subsets of the parameter space. (So if we are over a countable field, the locus of very general objects might be empty!)

Finally, generic is sometimes used as a synonym for these things, but it shouldn't be. (As I understand it, some of the problems caused by lack of rigour in algebraic geometry in the pre-Zariski--Weil world stemmed from using the word generic in an imprecise way.) In modern terms, the generic point $\eta$ of an irreducible variety means the unique point whose closure is the whole variety. Given a family of objects over the variety $\pi:U \rightarrow V$, one then gets the generic object in the family as the fibre of $\pi$ over the point $\eta$.

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