Hence, the axiom A2 of

affine plane we have that [[??].sub.RP.sup.P'] = R'P'.

Let X be the

affine plane k-curve defined by the datum of the polynomial F = [T.sup.3.sub.1] - [T.sup.2.sub.2] [member of] C[[T.sub.1], [T.sub.2]].

If the projective plane algebraic curve C is defined by the polynomial F(x, y, z), then the corresponding

affine plane algebraic curve C* is defined by the dehomogenization f(x, y) of F(x, y, z),

Blow up the

affine plane [A.sup.2] = Spec k[[e.sub.1], [e.sub.2]] = Spec k[Q] at the origin Y, and remove the strict transform of the divisor Z defined by [e.sub.2] = 0.

In order to teach students some of the most important techniques used for constructing combinatorial designs, Lindner and Rodger (both Auburn U.) focus on several basic designs: Steiner triple systems, latin squares, and finite projective and

affine plane. Having set these out, they start to add additional interesting properties that may be required, such as resolvability, embeddings, orthogonality, and even some more complicated structures such as Steiner quadruple systems.

Recall that a finite

Affine plane of order n must have [n.sup.2] points and [n.sup.2]+n lines, with n points to a line.

For example, Bezout's theorem, that the number of points of intersection of two algebraic curves is equal to the product of their degrees, would obviously suffer greatly if restricted to the real

affine plane."

This collection of [5.sup.2] = 25 words as "points" and [5.sup.2] + 5 = 30 "lines" of chess piece sweeps makes the square an order 5 finite

Affine plane. See Oscar Thumpbindle's article in this issue of Word Frays for further details.

The

affine plane associated to the Lorentzian vector space [L.sup.2] is called the Lorentzian plane; see, e.g., [3] and [20, [section] 11 and [section] 12].

This means that x describes the immersion of an

affine plane parallel to O[x.sup.1][x.sup.2].

If the partial spread is a spread, the translation net becomes an

affine plane of order [q.sup.t]; a 'translation plane' of order [q.sup.t].

In his proof he makes essential use of the fact that such circle planes sit in [R.sup.3] and that circles come from

affine planes of [R.sup.3].