Enumerative geometry of singular plane curves

general points in the plane? The answer is easy for 6 = 1 (namely 3 (d l ) 2) but otherwise was, to my knowledge, unknown until now. We are going to develop a recursive procedure for solving such problems. To make the recursion possible, it will be necessary to consider a generalization of independent interest of the above problem. Let S be the blow-up of IP 2 at a point P, with exceptional divisor E1 and a fixed line section Eo (we call S an irreducible fan). A curve C c S is said to be of type (d, e) if it is the proper transform of a curve of degree d in ~z having multiplicity e at P. One then asks: how many curves C c S of given type (d, e), having only smooth branches of given configuration along E0 w E1 and otherwise having a given number of nodes as their only singularities, pass through the "appropriate" number of points of S, some of which may be on E o w E~ ? We will solve this problem by recursion on d e . The recursion is based on degenerating the surface S to a surface (called a 2-fan) of the form So = $1 w S 2 with Si--S, i = i, 2. Using and suitably enhancing methods of [3], we can describe a suitable part of the limit on So of the relevant families of curves of type (d, e) on S, d > e + 1, in terms of similar families of curves of type (d,e') and (e', e), e ' < e = d on $1 and $2, respectively.