| In this thesis, we study the locus of intersecting cycles inside a product of Chow varieties of a smooth projective variety X. The main case considered is that of pairs of 1-cycles on a threefold. In this situation we construct a Cartier divisor supported on the incidence locus J↪C1 (X) x C1 (X). We also study the case J↪C0 (X) x CdimX -1 (X), and here we make use of explicit descriptions of both Chow varieties.;In both cases we proceed by defining an incidence line bundle L on a product of Hilbert schemes mapping to the corresponding Chow varieties. The essential ingredients of the incidence bundle are the universal families over the Hilbert schemes and the determinant line bundle of a perfect complex. We are thus led to problems of descent: to define an isomorphism between two pullbacks of L , satisfying the cocycle condition; and then to show the effectiveness of the descent datum thus obtained.;The first step towards defining the descent datum is a characterization of functions on a seminormal scheme as pointwise functions compatible with specialization. Along with a straightforward K-theoretic interpretation of the Hilbert-Chow morphism, this characterization converts the problem of defining the descent datum to understanding how K-theory behaves under specialization. As for the effectiveness, the seminormality of the Chow variety produces a criterion for effective descent, and the explicitness of the descent datum allows us to verify it in our situation. |
