Sulle rigate ellittiche

Lanteri, Antonio ; Palleschi, Marino

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 67 (1979) no. 1-2, pp. 87-94

Résumé

Let $X \subset \mathbf{P}^{r}$ be a complex smooth algebraic surface, the general hyperplane section of which is an elliptic curve. A classical Theorem due to G. Castelnuovo ([1]) states that if $X$ is not an elliptic scroll then $X$ is a rational surface. Castelnuovo achieves this result by showing that if $X$ is not a scroll, then a suitable linear system of hypersurfaces in $\mathbf{P}^{r}$ exhibits $X$ as a projective model of a surface of degree $d$ in $\mathbf{P}^{d}$, which is not a scroll; hence $X$ is rational. In this paper we supply a new proof of the previous result (Teorema 3.1) (over an algebraically closed field). This proof allows us to describe, in the class of the (smooth) linearly normal surfaces, the elliptic scrolls as the surfaces of degree $d$ in $\mathbf{P}^{d-1}$ with elliptic general hyperplane section. Our argument is supported by the following fact (Proposizione 3.1): let $\rho:S\rightarrow Y \subset \mathbf{P}^{r-1}$ be the projection of a smooth surface $S \subset \mathbf{P}^{r}$ from a point $p \in S$; if $Y$ is a (smooth) scroll then either $S$ is a rational surface or $S$ itself is a scroll. In the latter case $\rho$ is an elementary transformation with center $p$; hence the elementary transformations, introduced by Nagata in [5], can be seen as projections. Finally we give an explicit description of projective models of the elliptic scrolls. This construction generalizes the one given in [3] for the elliptic scroll in $\mathbf{P}^{4}$ and shows the links between the degree, the invariant and the hyperplane class of such surfaces.