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).
Reviewer: E.Arrondo (Madrid)


14J70 Hypersurfaces and algebraic geometry
14N05 Projective techniques in algebraic geometry
14J30 \(3\)-folds
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