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On a conjecture of Clemens on rational curves on hypersurfaces. (English) Zbl 0883.14022
In Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 629-636 (1986; Zbl 0611.14024)], H. Clemens conjectured that a general hypersurface $$X$$ in $${\mathbb{P}}^n$$, $$n\geq 4$$, of degree $$d\geq 2n-2$$ contains no rational curve, and he proved the result for $$d\geq 2n-1$$. In this paper the author proves the following result, which implies Clemens’ conjecture:
Let $$X \subset {\mathbb{P}}^n$$ be a general hypersurface of degree $$d\geq 2n-l-1,$$ with $$1\leq l\leq n-3$$; then any subvariety $$Y$$ of $$X$$ of dimension $$l$$ has a desingularization $$\tilde{Y}$$ with an effective canonical bundle; if the inequality is strict, the sections of $$K_{\tilde{Y}}$$ separate generic points of $$Y$$ (of course, for $$l=1$$, $$Y$$ cannot be rational since in that case $$K_{\tilde{Y}} \cong {\mathcal O}_{\tilde{Y}}(-2)$$ which is not effective).
Notice that hypersurfaces of degree $$\leq 2n-3$$ do contain lines, so in this sense this result is optimal.
The paper is based on techniques developed by H. Clemens and L. Ein, based on the study of the bundle $$T{\mathcal X}(1)_{\mid X}$$, where $${\mathcal X}$$ is the universal family of hypersurfaces in $${\mathbb{P}}^n \times H^0({\mathcal O}_{{\mathbb{P}}^n}(d))^0$$ , where the last factor is the open set of smooth hypersurfaces. The author can improve the previous results since she manages to study sections of $$\bigwedge ^2 T{\mathcal X}(1)_{\mid X}$$ instead of those of $$\bigwedge ^2 T{\mathcal X}(2)_{\mid X}$$, as it was done before.

MSC:
 14J70 Hypersurfaces and algebraic geometry 14H45 Special algebraic curves and curves of low genus 14M10 Complete intersections 14M20 Rational and unirational varieties
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