On a conjecture of Clemens on rational curves on hypersurfaces.

*(English)*Zbl 0883.14022In 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.

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.

Reviewer: A.Gimigliano (Firenze)