The Hilbert-Chow morphism and the incidence divisor: zero-cycles and divisors
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 \({\mathcal U}_b\) are disjoint. One expects that, for \(a+b = n-1\), \(U\) is dense. Since \(\text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) is acyclic on \(U\), the ``Div'' construction of Knudsen and Mumford shows that \(\mathcal L\) is the invertible sheaf associated to a canonically defined Cartier divisor on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\). One faces, now, the problem of showing that \(\mathcal L\) descends to a line bundle on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) (via the product of the Hilbert-Chow morphisms). In order to solve this problem, the author studies the morphism of Picard groups induced by a seminormal proper hypercovering of a seminormal scheme, using some results of \textit{L. Barbieri-Viale} and \textit{V. Srinivas} [``Albanese and Picard 1-motives'', Mém. Soc. Math. Fr., Nouv. Sér. 87 (2001; Zbl 1085.14011)]. Then, by analysing the Hilbert-Chow morphism for 0-cycles, using the results of [\textit{B. Iversen}, Linear determinants with applications to the Picard scheme of a family of algebraic curves.Lecture Notes in Mathematics. 174. Berlin-Heidelberg-New York: Springer-Verlag. (1970; Zbl 0205.50802)], and for codimension 1 cycles, using the results of Knudsen and Mumford, the author shows that, in the case \(a=0\), \(b = n-1\), \(\mathcal L\) descends from \({\mathcal H}_0(X)\times {\mathcal H}_{n-1}(X)\) to \({\mathcal C}_0(X)\times {\mathcal C}_{n-1}(X)\).
scientific article; zbMATH DE number 5960888
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