Abstract. Let X, Y be smooth varieties of dimensions n, n + 1 over an algebraically closed field, and f:X → Y a finite map, birational onto its image Z. The source double-point set supports two natural positive cycles: (1) the fundamental cycle of the divisor M2 denned by the conductor of X/Z, and (2) the direct image of the fundamental cycle of the residual scheme X2 of the diagonal in the product X XyX. Over thirteen years ago, it was conjectured that the two cycles are equal if the characteristic is 0 or f is “appropriately generic.” That conjecture will be established in a more general form.

Introduction

Let X and Y be smooth varieties over an algebraically closed field, and assume that dim Y − dim X = 1. Let f:X → Y be a finite map that is birational onto its image Z. For example, X might be a projective variety, and f a general central projection onto a hypersurface Z in Y := Pn+1. Consider the source double-point scheme M2 of f. By definition, M2 is the effective divisor whose ideal is the conductor Cx of X/Z. Its underlying set consists of the points x of X whose fiber f−1f(x) is a scheme of length at least 2. Consider also the residual scheme X2 of the fiber product X xyX with respect to the diagonal. By definition, X2 := P(I(Δ)) where I(Δ) is the ideal of the diagonal. Consider finally the map f1:X2 → X induced by the second projection.