The Hilbert-Chow morphism and the incidence divisor: zero-cycles and divisors
type: Item
instance of: scholarly article
title: The Hilbert-Chow morphism and the incidence divisor: zero-cycles and divisors
zbMATH Open document ID: 1231.14003
author: Joseph Ross
published in: Journal of Pure and Applied Algebra
publication date: 2011-10-21T00:00:00Z
review text: Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \({\mathcal C}_d(X)\) be the \textit{Chow variety} parametrizing \(d\)-dimensional cycles on \(X\) (see, for example, the book of [\textit{J. Kollár}, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag (1995; Zbl 0877.14012)] for its construction). For \(a+b = n-1\), let \({\mathcal I}\subset {\mathcal C}_a(X)\times {\mathcal C}_b(X)\) be the incidence variety parametrizing the pairs \((A,B)\) with \(A\cap B \neq \emptyset\). When \(k = {\mathbb C}\), B. Mazur constructed, in 1993, using intersection theory operations on the universal cycles, a Weil divisor on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) supported on \(\mathcal I\) and posed the problem whether \(\mathcal I\) is the support of a Cartier divisor, satisfying some additional properties. \textit{B. Wang} [Compos. Math. 115, No. 3, 303--327 (1999; Zbl 0982.14017)] showed that the Weil divisor \((n-1)!\, {\mathcal I}\) is Cartier. In the paper under review, the author proposes a new approach to Mazur's question. Let \({\mathcal H}_d(X)\) be the \textit{Hilbert scheme} parametrizing \(d\)-dimensional subschemes of \(X\). Let \({\mathcal U}_a\), \({\mathcal U}_b\) be the closed subschemes of \(X\times {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) obtained by pulling back the universal families over \({\mathcal H}_a(X)\) and \({\mathcal H}_b(X)\). Using the \textit{determinant functor} constructed by \textit{F. Knudsen} and \textit{D. Mumford} [Math. Scand. 39, No. 1, 19--55 (1976; Zbl 0343.14008)], one gets a line bundle \({\mathcal L} := \text{det}\, \text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\). Let \(U\subset {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) be the open subset over which the fibers of \({\mathcal U}_a\) and \(…
reviewed by: Iustin Coandă
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 5960888
zbMATH Keywords: Chow variety
zbMATH Keywords: incidence divisor
zbMATH Keywords: Hilbert-Chow morphism
zbMATH Keywords: determinant functor
zbMATH Keywords: seminormal variety
zbMATH Keywords: simplicial Picard functor
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.03.001
OpenAlex ID: W2023857897
cites work: Q3929858
cites work: Albanese and Picard 1-motives
cites work: Q4097533
cites work: INCIDENCE DIVISOR
cites work: Néron Models
cites work: Truncated Hilbert functors.
cites work: Q4200313
cites work: Éléments de géométrie algébrique. I: Le langage des schémas. II: Étude globale élémentaire de quelques classe de morphismes. III: Étude cohomologique des faisceaux cohérents (première partie)
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Q4143433
cites work: Brauer group of a linear algebraic group
cites work: The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div".
cites work: Q4859907
cites work: Q4740026
cites work: Lois polynomes et lois formelles en théorie des modules
cites work: The Hilbert-Chow morphism and the incidence divisor
cites work: A characterization of seminormal schemes
cites work: Q4238567
DOI: 10.1016/J.JPAA.2011.03.001
Recommended article: The Hilbert-Chow morphism and the incidence divisor
Recommended article: Q4763360
Recommended article: Intersections of Hirzebruch--Zagier divisors and CM cycles
Recommended article: On the Hilbert function of zero-dimensional schemes in \(\mathbb P^{1} \times \mathbb P^{1}\)
Recommended article: Divisorial Zariski decomposition and algebraic Morse inequalities
Recommended article: Some remarks on the divisorial Zariski decomposition on varieties whose first Chern class is zero
Recommended article: On the Chow group of zero-cycles of a generalized Kummer variety
Recommended article: Q4883842
Recommended article: Homology of zero-divisors
Recommended article: The arithmetic of the Chow group of zero-cycles
description: scientific article; zbMATH DE number 5960888