is 6H, and Steiner calculated (6H)5 = 65P as H5=P (where P is the class of a point in the Chow ring). However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E. Suppose that L = 2H−E is the strict transform of the conics tangent to a given line. Then the intersection numbers of H and L are given by H5=1P, H4L=2P, H3L2=4P, H2L3=4P, H1L4=2P, L5=1P. So we have (6H−2E)5 = (2H+2L)5 = 3264P. gave a precise description of exactly what "general position" means (although their two propositions about this are not quite right, and are corrected in a note on page 29 of their paper). If the five conics have the properties that there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a "double line with 2 marked points" tangent to all 5 conics) no three of the conics pass through any point (otherwise there is a "double line with 2 marked points" tangent to all 5 conics passing through this triple intersection point) no two of the conics are tangent no three of the five conics are tangent to a line a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics) then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. Here the multiplicity is given by the product over all 5 conics Ci of (4 − number of intersection points of C and Ci). In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics. Over other algebraically closed fields the answer is similar, unless the field has characteristic 2 in which case the number of conics is 51 rather than 3264. References External links Intersection theory Algebraic geometry