Blocking set

Abstract: A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let

Abstract: 1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, hence order q = pn , where p is an odd prime and h is a positive integer. It is well known that every straight line and every non-singular conic of then contains q + 1 points exactly.

Abstract: We show that the Yang-Mills instantons can be described in terms of certain holomorphic bundles on the projective plane. The proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries.