Degree of the fibres of an elliptic fibration

Alexandru Buium

Annales de l'institut Fourier (1983)

  • Volume: 33, Issue: 1, page 269-276
  • ISSN: 0373-0956

Abstract

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Let X → B an elliptic fibration with general fibre F . Let n e , n s , n a , n v be the minima of the non-zero intersection numbers ( ℒ , F ) where ℒ runs successively through the following sets: effective divisors on X , invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let m be the maximum of the multiplicities of the fibres of X → B . We prove that n e = n s if and only if n e ≥ 2 m and that n a = n v if and only if n a ≥ 3 m .

How to cite

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Buium, Alexandru. "Degree of the fibres of an elliptic fibration." Annales de l'institut Fourier 33.1 (1983): 269-276. <http://eudml.org/doc/74572>.

@article{Buium1983,
abstract = {Let $X\rightarrow B$ an elliptic fibration with general fibre $F$. Let $n_e,n_s,n_a,n_v$ be the minima of the non-zero intersection numbers $(\{\cal L\},F)$ where $\{\cal L\}$ runs successively through the following sets: effective divisors on $X$, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X\rightarrow B$. We prove that $n_e=n_s$ if and only if $n_e\ge 2m$ and that $n_a=n_v$ if and only if $n_a\ge 3m$.},
author = {Buium, Alexandru},
journal = {Annales de l'institut Fourier},
keywords = {elliptic fibration; intersection numbers; effective divisors; very ample divisors; invertible sheaves},
language = {eng},
number = {1},
pages = {269-276},
publisher = {Association des Annales de l'Institut Fourier},
title = {Degree of the fibres of an elliptic fibration},
url = {http://eudml.org/doc/74572},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Buium, Alexandru
TI - Degree of the fibres of an elliptic fibration
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 1
SP - 269
EP - 276
AB - Let $X\rightarrow B$ an elliptic fibration with general fibre $F$. Let $n_e,n_s,n_a,n_v$ be the minima of the non-zero intersection numbers $({\cal L},F)$ where ${\cal L}$ runs successively through the following sets: effective divisors on $X$, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X\rightarrow B$. We prove that $n_e=n_s$ if and only if $n_e\ge 2m$ and that $n_a=n_v$ if and only if $n_a\ge 3m$.
LA - eng
KW - elliptic fibration; intersection numbers; effective divisors; very ample divisors; invertible sheaves
UR - http://eudml.org/doc/74572
ER -

References

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  1. [1] A. BEAUVILLE, Surfaces algébriques complexes, Astérisque, 54 (1978). Zbl0394.14014MR58 #5686
  2. [2] E. BOMBIERI, Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1972), 171-220. Zbl0259.14005MR47 #6710
  3. [3] F. ENRIQUES, Le superficie algebriche, Zanichelli, 1949. Zbl0036.37102MR11,202b
  4. [4] P. GRIFFITHS and J. HARRIS, Principles of algebraic geometry, John Wiley & Sons, New York, 1978. Zbl0408.14001MR80b:14001
  5. [5] K. KODAIRA, On compact complex analytic surfaces II, Ann. of Math., 77 (1963). Zbl0118.15802MR29 #2822
  6. [6] C.P. RAMANUJAM, Remarks on the Kodaira vanishing theorem, J. of the Indian Math. Soc., 36 (1972), 41-51 ; Supplement to the article “Remarks on the Kodaira vanishing theorem”, J. of the Indian Math. Soc., 38 (1974), 121-124. Zbl0276.32018MR48 #8502

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