Curves on generic hypersurfaces. (English) Zbl 0611.14024

Clemens, Herbert

Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, No. 4, 629-636 (1986); erratum ibid. 20, No. 2, 281 (1987).

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

Keywords:

hypersurface; algebraic family of immersed curves; codimension