## Subvarieties of general hypersurfaces in projective space.(English)Zbl 0823.14030

The well-known Noether-Lefschetz theorem states that any curve on a general surface $$S$$ of degree $$d\geq 4$$ in $$\mathbb{P}^ 3$$ is the complete intersection of $$S$$ with another surface of degree $$k$$. The main result in this paper is that, if $$d\geq 5$$, the general $$S$$ does not possess curves of geometric genus $$g\leq {1\over 2} d(d-3)-3$$. This improves a conjecture by Harris and a previous result of Clemens. Moreover, the bound given by the author is sharp since it is achieved by the tritangent plane sections; the author also proves that if $$d\geq 6$$ then these are the only curves achieving the bound. The general idea of this proof is to study the possible singularities that could possess the complete intersection of $$S$$ with a surface of degree $$k$$; thus the author gets a bound $$g\geq {1\over 2} dk(d-5) +2$$. Then the result follows at once by just analyzing the case $$k=1$$.
A similar result is obtained in higher dimension. More precisely the author gives a lower bound for the geometric genus of any reduced and irreducible divisor $$M$$ on a general hypersurface of $$\mathbb{P}^{n+1}$$; moreover there is a bound $$C(n)$$ (which can be computed explicitly for each $$n$$) such that if $$d\geq C(n)$$ the bound is sharp and achieved only for hyperplane sections. The particular result for $$n=3$$ states that on a general 3-fold in $$\mathbb{P}^ 4$$ of degree $$d\geq 6$$ an integral divisor $$M$$ has geometric genus at least two. It is an open question posed by Green whether a quintic 3-fold of degree 5 in $$\mathbb{P}^ 4$$ could have surfaces with geometric genus zero (a negative answer would imply Clemens conjecture that such a 3-fold has only a finite number of rational curves of each degree).
 14J70 Hypersurfaces and algebraic geometry 14N05 Projective techniques in algebraic geometry 14J30 $$3$$-folds