What does «generic» mean in algebraic geometry?

Question

As a beginner, when I read some books in algebraic geometry such as the book complex projective variety by Mumford,I found a lot of "generic" object. Could any one tell me how to understand "generic"?

Answer 1

An irreducible scheme $B$ has a unique generic point $\eta$. The generic fiber of a family $X\to B$ is the fiber $X_{\eta}$ over that special point $\eta$.

A general fiber $X_b$ is a fiber over $b\in B$ that belongs to some fixed open set $U\subset B$. And very general means that $b$ belongs to $V$ which is a complement of countably many Zariski closed proper subsets $Z_i$ of $B$.

That is the most common modern terminology. In older (and not so old) books sometimes generic is used where general would be more appropriate.

Added in response to Kevin Lin's comment: In classical alg. geometry, people care about general fibers. The scheme theory provides generic fibers, which are really very convenient to have, since they are so concrete. The way "generic to general" usually works is as follows: You prove that the generic fiber has a property P, and that the property P is constructible. Then P holds for any $b$ in an open neighborhood of $\eta$, that is for a general $b$. EGAs contain a long list of properties which are constructible in proper (e.g. projective) families: smoothness, CM, normality, etc., etc.

(And, yes, similar things were discussed in multiple other MO questions. One thing MO seriously lacks is a clear organization of the accumulated knowledge, so that people do not constantly ask and answer variations of the same question.)

Answer 2

It usually means "belonging to a non-empty Zariski open (and hence, usually, dense) subset which can be made precise but we won't bother to do so" or some small variation of that. Usually, the specific open set in question is the one of those objects satisfying all the conditions imposed in the proof of whatever statement you are proving.

Of course, this is not a rule, and mathematicians have used 'generic' to mean all sort of different things...

Answer 3

Generic often refers to true in a Zariski open dense set, i.e. true outside some "small" set of proper codimension (in the Zariski topology). It is something like the algebro-geometric version of "almost everywhere" or "residual" in analysis and topology.

There is a general fact in algebraic geometry, already mentioned in previous answers, that whenever a constructible set (i.e., one obtained from closed sets using a finite number of boolean operations, at least one when has suitable noetherian hypotheses) contains the "generic point" of an irreducible scheme, it is generic in the sense of containing a Zariski open (and thus dense, under irreducibility hypotheses) set. One simple example of this phenomenon is the following: let $M$ be a matrix with coefficients in $k(T)$ for $k$ a field. Then the rank of $M$ ("at the generic point") is equal to the rank of almost all the specializations $M(t)$ obtained by substituting $T \to t, t \in k$ (i.e. in a "generic set").