Point in general position
A point on an algebraic variety that belongs to an open and dense subset $S$ in the Zariski topology. In algebraic geometry a point in general position is often called simply a generic point.
More precisely, a point is said to be in general position if it is outside a certain (given, or to be described) closed set; a point in an irreducible set $X$ is called generic if it is outside every closed set different from $X$ itself.
In differential topology, the phrase "a point in general position" often is used in the sense of a generic point, which is roughly "a point with no particular relationship of current importance to other structure elements being considered" . The precise meaning depends on the context. An element generically has a certain property if the property holds outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double roots. Two submanifolds $A,B\subset N$ are in general position if they "intersect as little as possible" . If $\dim A+\dim B<\dim N$, this means $A\cap B=\emptyset$; if $\dim A+\dim B\geq\dim N$, then for every $x\in A\cap B$, $T_xA+T_xB=T_xN$. Any non-general position situation can be changed to a general position situation by an arbitrarily small change, while if things are in general position, then sufficiently small changes do not change that. A precise version of general position is transversality, cf. also Transversality condition.
|[a1]||R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001|
|[a2]||M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 MR0448362 Zbl 0356.57001|
|[a3]||D.R.J. Chillingworth, "Differential topology with a view to applications" , Pitman (1976) pp. 221ff MR0646088 Zbl 0336.58001|
Point in general position. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Point_in_general_position&oldid=33239