General Position - More Generally

More Generally

This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections). In algebraic geometry this kind of condition is frequently encountered, in that points should impose independent conditions on curves passing through them.

For example, five points determine a conic, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.

General position is preserved under biregular maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the Veronese map is biregular; as points under the Veronese map corresponds to evaluating a degree d polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.

The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for higher degree.

This is not sufficient, however. While nine points determine a cubic, there are configurations of nine points that are special with respect to cubics, namely the intersection of two cubics. The intersection of two cubics, which is points (by Bézout's theorem), is special in that nine points in general position are contained in a unique cubic, while if they are contained in two cubics they in fact are contained in a pencil (1-parameter linear system) of cubics, whose equations are the projective linear combinations of the equations for the two cubics. Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the Cayley–Bacharach theorem that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.

For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces.

Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any k distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.