Subvarieties of generic complete intersections. (English) Zbl 0701.14002

The paper studies the Hilbert scheme of a generic complete intersection X of type \((m_ 1,...,m_ k)\) in the Grassmann variety G of r-planes in an \((n+1)\)-dimensional vector space \(V.\) If \(H_ 0\) is an open irreducible set of the Hilbert scheme, which parametrizes smooth irreducible subvarieties of X, let \({\mathcal Z}_ 0\to H_ 0\) be the corresponding universal family of subvarieties, let F: \({\mathcal Z}_ 0\to X\) be the natural map, and let \(Y=F({\mathcal Z}_ 0)\). The main result of the paper shows for a general member of the family \(H_ 0\) that \(text{codim}_ XY\geq m_ 0+m_ 1+...+m_ k-n-1,\) where \(m_ 0\) is the least integer such that \(H^ 0(K_ Z\otimes {\mathcal O}_ Z(m_ 0))\neq 0\), and that \(N_{Z/X}\otimes {\mathcal O}_ Z(1)\) is generated by its sections. A version of the argument establishes the following theorem:
If \(H_ 0\) parametrizes smooth curves of degree \(d\) and genus \(g\) in X, and C is a general curve of \(H_ 0\) such that C spans an \(n_ 0\)- dimensional space in \(P(\bigwedge^ rV)\), then \(co\dim_ XY\geq (2- 2g+(m-n-1)d)/(d-n_ 0+1).\) This generalizes a result of H. Clemens [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 629-636 (1986) and 20, 281 (1987; Zbl 0611.14024 and Zbl 0625.14012)] which treats the case of curves on generic hypersurfaces.
Reviewer: L.L.Avramov


14C05 Parametrization (Chow and Hilbert schemes)
14M10 Complete intersections
14M15 Grassmannians, Schubert varieties, flag manifolds
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