TITLE: What  does «generic» mean in algebraic geometry?
QUESTION [13 upvotes]: 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"?

REPLY [4 votes]: 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").