The source double-point cycle of a finite map of codimension one

Summary

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 X_yX. 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 C_x 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 x_yX 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.