The multiple-point schemes of a finite curvilinear map of codimension one

Abstract

Let X and Y be smooth varieties of dimensions n−1 and n over an arbitrary algebraically closed field, f : X → Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, for all x ∈ X, the Jacobian ∂f (x) has rank at least n−2. For r ≥ 1, consider the subscheme N r of Y defined by the (r − 1)-th Fitting ideal of the O Y -module f * O X , and set M r := f −1 N r . In this setting-in fact, in a more general setting-we prove the following statements, which show that M r and N r behave like reasonable schemes of source and target r-fold points of f .If each component of M r , or equivalently of N r , has the minimal possible dimension n − r, then M r and N r are Cohen-Macaulay, and their fundamental cycles satisfy the relation, f * [M r ] = r[N r ]. Now, suppose that each component of M s , or of N s , has dimension n − s for s = 1, . . . , r + 1. Then the blowup Bl(N r , N r+1 ) is equal to the Hilbert scheme Hilb r f , and the blowup Bl(M r , M r+1 ) is equal to the universal subscheme Univ r f of Hilb r f × Y X; moreover, Hilb r f and Univ r f are Gorenstein. In addition, the structure map h: Hilb r f → Y is finite and birational onto its image; and its conductor is equal to the ideal J r of N r+1 in N r , and is locally self-linked. Reciprocally, h * O Hilb r f is equal to Hom(J r , O N r ). Moreover, h * [h −1 N r+1 ] = (r + 1)[N r+1 ]. Similar assertions hold for the structure map h 1 : Univ r f → X if r ≥ 2.1980 Mathematics Subject Classification (1985 Revision). Primary 14C25; Secondary 14O20, 14N10.Acknowledgements. It is a pleasure to thank Luchezar Avramov, Winfried Bruns, David Eisenbud, Hans-Bjørn Foxby, and Christian Peskine for fruitful discussions. Avramov and Foxby discussed Gorenstein maps; Bruns and Eisenbud discussed properties of Fitting ideals; and Peskine explained at length his work [12] with Gruson.

Fitting ideals and multiple points of surface parameterizations

“…Given a finite map of schemes ϕ : X → Y of codimension one, its source and target double-point cycles have been extensively studied in the field of intersection theory (see e.g. [KLU96,KLU92,Pie78,Tei77]). In [MP89], Fitting ideals are used to give a scheme structure to the multiple-point loci.…”

Разрешение Особенностей Коранга $1$ Фронта Общего Положения

“…Эта конструкция использовалась в комплексных задачах при исследовании циклов кратных точек общих голоморфных отображений коранга 1 (см. [9,10,[12][13][14]). Однако, в отличие от принципа итерации, мы рассматриваем C ∞ -гладкие лежандровы отображения и изучаем циклы произвольных устойчивых мультиособенностей коранга 1.…”

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

“…However, the latter is equal to F 2 by Proposition 3.6(3). [22,Lemma 2.3], and if f is birational onto its image and Y satisfies (S 2 ), then I N2 has grade at least 2 by Lemma 2.11. So the assertions follow from Proposition 3.8.…”