Ranks of Webs and Nets

G. Eric Moorhouse

We are currently (January, 2005) using the University of Wyoming ISC Beowulf Linux cluster Euler to enumerate nets of order 11 and large 11-rank, comparable to our lists for nets of order 5 and 7; see definitions below. These computations may play a role in a future determination of projective planes of order 11. (The enumeration of planes of order 11 by methods used in the computer enumeration of projective planes of order 9 and 10 is infeasible given current computational resources.)
Sophus Lie (1842-1899)

A recurrent theme in the work of Sophus Lie was that of a double translation surface. Any two curves C1 and C2 in n-space Fn generate a translation surface C1+C2 consisting of the set of points of the form v1+v2 where vi is a point of Ci.

A surface which is expressible in two different ways as a sum of two curves, namely as C1+C2=C3+C4, is a double translation surface.

Example: F is an arbitrary field;

C1={(x, 0, 2x2) : x in F},

C2={(0, y, –y2) : y in F},

C3={(s, 2s, –2s2) : s in F},

C4={(t, t, t2) : t in F};

the surface C1+C2=C3+C4 is given by z=2x2–y2.

Lie's original motivation for studying such surfaces was that they frequently arose as examples of minimal surfaces; but connections to other mathematical concepts soon became apparent. Lie proved that for a double translation surface as above, all four curves and the resulting surface lie in an affine subspace of dimension at most 3, and the tangent lines to all four curves meet the plane at infinity in an algebraic curve of degree four. Conversely, an algebraic curve of degree 4 and arithmetic genus 3 in the plane at infinity, gives rise to an example of such a surface. This version of the theorem refers to the case F is the complex numbers, and the curves Ci are complex analytic; a version of the theorem for real smooth curves is also known.

Let F be either the real numbers or the complex numbers. A k-web (over F) is a smooth surface (i.e. 2-manifold) S over F having k smooth ‘coordinate’ functions ui : S –> F (i=1,2,...,k) such that for any distinct i and j, every point P of S is uniquely determined by the coordinate pair (ui(P),uj(P)). We also assume that the level curves of ui meet the level curves of uj transversely (i.e. level curves for different coordinates have no common tangent). We may assume that the origin 0 of Fn is a point of S, and that ui(0)=0 for all i. The rank of the web is the dimension of the vector space consisting of all k-tuples (f1,f2,...,fk) of smooth functions F –> F such that fi(0)=0 and f1(u1(P))+f2(u2(P))+...+fk(uk(P))=0 for every point P of S. A 4-web is the same thing as a double translation surface of rank (either 2 or 3) equal to the dimension of the affine subspace generated by the surface.
The finite analogue of a web is a net. A k-net of order n is an incidence system of n2 points and nk lines such that the lines form k parallel classes of n lines each; each line has n points; and each point lies on k lines, one from each parallel class. Here two lines are said to be parallel if they are either equal or disjoint; two non-parallel lines meet in exactly one point. We must have k<=n+1, and an (n+1)-net of order n is the same thing as an affine plane of order n, which in turn is equivalent to a projective plane of order n having a distinguished line. At the right is shown is an affine plane of order 3, i.e. a 4-net of order 3.
Fix a prime p dividing n, and let F be a field of characteristic p. (The integers mod p will suffice.) We assume that p divides n only once. By usual convention, the p-rank of a net is the rank of its point-line incidence matrix A of size n2-by-nk; however, consistency with the infinite case described above, requires that we refer to the p-rank of such a net as the null(A)–k+1. We conjecture that this value is at most (k–1)(k–2)/2, which is known to be the upper bound for the rank of a k-web over the real or complex numbers. Any plane of order n must contain nets with high p-rank. It is reasonable to hope that this fact may greatly facilitate the enumeration of planes of order n for certain small values of n.