On the normal bundles of rational space curves
The normal bundle of rational curves in a projective space has been studied, in the last few years, by \textit{D. Eisenbud} and \textit{A. Van de Ven} [Math. Ann. 256, 453-463 (1981; Zbl 0443.14015)] and by \textit{F. Ghione} and \textit{G. Sacchiero} [Manuscr. Math. 33, 111-128 (1980; Zbl 0496.14021)]. The first two authors dealt with characteristic zero, proving that, if \(\phi:\quad {\mathbb{P}}^ 1\to {\mathbb{P}}^ 3\) is a closed immersion, with \(\phi^*{\mathcal O}_{{\mathbb{P}}^ 3}(1)={\mathcal O}_{{\mathbb{P}}^ 1}(d),\) then the normal bundle \(N_{\phi}\) of \(\phi\), namely the dual of the kernel of the natural map \(\phi^*:\quad \Omega^ 1_{{\mathbb{P}}^ 3}\to \Omega^ 1_{{\mathbb{P}}^ 1},\) splits in the following way: \(N_{\phi}={\mathcal O}_{{\mathbb{P}}^ 1}(2d-1+\rho)\oplus {\mathcal O}_{{\mathbb{P}}^ 1}(2d-1-\rho),\) where \(0\leq \rho \leq d-4\). Ghione and Sacchiero examined the same question in the more general setting: \(\phi\) not ramified and the characteristic p of the base field not a divisor of d. Their result says that \(0\leq \rho \leq d-1\); further \(\rho =d-1\) if and only if \(\phi ({\mathbb{P}}^ 1)\) is a plane curve, \(\rho =d-2\) cannot happen and the curves with \(\rho =d-3\) are completely classified. The author studies the problem of the splitting of \(N_{\phi}\) in the cases not covered by Ghione and Sacchiero's analysis, namely when p divides d. His approach follows, with suitable changes, Ghione and Sacchiero's one, and the results are the same for \(p\neq 2\). When \(p=2\) he shows, correcting an oversight by Gione and Sacchiero, that there are rational curves with \(\rho =d-3\) and gives a very simple necessary and sufficient condition for such curves to exist. Again for \(p=2\) the following is proved: the possible values for \(\rho\) are the even numbers between 0 and d-1 if d is odd, the odd numbers between 0 and d-1 if d is even. This contrasts with the case \(p=0\). Finally a few examples are analysed, in which the splitting of \(N_{\phi}\) can be realised with line bundles with a simple geometrical meaning.
scientific article; zbMATH DE number 3946338
scientific article