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


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