Symmetric Sets of Curves and Combinatorial Arrays

Let V be an algebraic curve. P is a set of points on V . C is a set of curves each of which intersects V at some points of P. We denote IP (C, V ) as the intersection multiplicity of C ∈ C with V at the point P ∈ P. If (V,P, C) satisfies the following conditions, we call it a symmetric set of curves: (1) for any point p ∈ P, the number of C having intersection multiplicity a is λa, (2) for any ordered pair (p, q) points of P, the number of curves C ∈ C is equal to λa,b satisfying Ip(C, V ) = a and Iq(C, V ) = b. When we arrange the multiplicities in an array of size |C| × |P|, we have a combinatorial array called a balanced array. We show a general construction of a symmetric set of curves using the Riemann-Roch theorem. We also consider a case that V is an elliptic curve and C is a set of conics. We show, in this case, the problem is reduced to find a set P of points which is the intersection of V and a curve of degree 4 derived from V .