## 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

### MSC:

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

### Citations:

Zbl 0611.14024; Zbl 0625.14012
Full Text:

### References:

 [1] Clemens, H.: Curves on generic hypersurfaces. Ann. Sci. Ec. Norm. Super.19, 629-636 (1986) · Zbl 0611.14024 [2] Clemens, H.: On subvarieties of intersection manifolds of projective manifolds (Preprint) [3] Ein, L.: Hilbert scheme of smooth space curves. Ann. Sci. Ec. Norm. Super.19, 469-478 (1986) · Zbl 0606.14003 [4] Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo and equations defining space curves. Invent. Math.72, 491-506 (1983) · Zbl 0565.14014 [5] Okonek, C., Schneider, M., Spindler, H.: Vector bundles a complex projective space. Progr. Math. 3. Basel, Boston: Birkhäuser, 1986 · Zbl 0438.32016
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