Sulle rigate ellittiche

Antonio Lanteri; Marino Palleschi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1979)

  • Volume: 67, Issue: 1-2, page 87-94
  • ISSN: 0392-7881

Abstract

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Let X ⊂ 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 𝐏 r exhibits X as a projective model of a surface of degree d in 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 𝐏 d - 1 with elliptic general hyperplane section. Our argument is supported by the following fact (Proposizione 3.1): let ρ : S → Y ⊂ 𝐏 r - 1 be the projection of a smooth surface S ⊂ 𝐏 r from a point p ∈ S ; if Y is a (smooth) scroll then either S is a rational surface or S itself is a scroll. In the latter case ρ 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 𝐏 4 and shows the links between the degree, the invariant and the hyperplane class of such surfaces.

How to cite

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Lanteri, Antonio, and Palleschi, Marino. "Sulle rigate ellittiche." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 67.1-2 (1979): 87-94. <http://eudml.org/doc/288846>.

@article{Lanteri1979,
author = {Lanteri, Antonio, Palleschi, Marino},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
language = {ita},
month = {7},
number = {1-2},
pages = {87-94},
publisher = {Accademia Nazionale dei Lincei},
title = {Sulle rigate ellittiche},
url = {http://eudml.org/doc/288846},
volume = {67},
year = {1979},
}

TY - JOUR
AU - Lanteri, Antonio
AU - Palleschi, Marino
TI - Sulle rigate ellittiche
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1979/7//
PB - Accademia Nazionale dei Lincei
VL - 67
IS - 1-2
SP - 87
EP - 94
LA - ita
UR - http://eudml.org/doc/288846
ER -

References

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  1. Castelnuovo, G. (1894) - Sulle superficie algebriche le cui sezioni piane sono curve ellittiche. «Rend. Accad. Naz. Lincei» (5) 3, 229-232. MR330515
  2. HartShorne, R. (1977) - Algebraic Geometry. Springer Verlag, Berlin-Heidelberg-New York. Zbl0367.14001
  3. Lanteri, A. e Palleschi, M. (1978) - Osservazioni sulla rigata geometrica ellittica di 𝐏 4 . «Istituto Lombardo (Rend. Sc.)», A 112, 223-233. MR463157
  4. Lanteri, A. e Palleschi, M. (1979) - Sulle superfici di grado piccolo in 𝐏 4 . «Istituto Lombardo (Rend. Sc.)», A 113 (in corso di stampa). 
  5. Nagata, M. (1960) - On rational surfaces I. «Mem. Coll. Sci. Kyoto» (A) 32, 351-370. 
  6. Nagata, M. (1970) - On self-intersection number of a section on a ruled surface. «Nagoya Math. J.», 37191-196. MR126444
  7. Shafarevich, I.R. e altri (1965) - Algebraic Surfaces. «Proc. Steklov Inst. Math.», 75 (trad. «Amer. Math. Soc.», 1967). MR258829

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