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Curves on generic hypersurfaces. (English) Zbl 0611.14024
Let $$V\subset {\mathbb{P}}^ n$$ be a smooth hypersurface of degree $$m\geq 2$$. An immersed curve $$f: C\to V$$ is defined to be a morphism which is everywhere of maximal rank from a complete nonsingular curve C. First of all, the author proves that for generic hypersurfaces W of degree m in $${\mathbb{P}}^{n+m}$$ such that $$W\cap {\mathbb{P}}^ n=V$$, the normal bundle $$N_{f,w}=f^*(T_ W)/T_ C$$ is semipositive. Let $${\mathcal G}$$ be an irreducible algebraic family of immersed curves of genus $$g$$ on V which covers a quasi-projective variety of codimension D in V. By extending $${\mathcal G}$$ to a family $${\mathcal F}$$ on W, and showing the semipositivity of certain vector bundles induced from the image in $$H^ 0(N_{f,W})$$ of the Kodaira-Spencer map at $$f\in {\mathcal G}\subseteq {\mathcal F}$$, the author finally proves the following theorem:
If V is generic, then $$D\geq (2-2g)/\deg(f)+m-(n+1)$$.
Reviewer: N.Nakayama

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14M07 Low codimension problems in algebraic geometry 14J99 Surfaces and higher-dimensional varieties
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