Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

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 $g$.

share|improve this answer

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 $\eta$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.