Poincaré bundles for projective surfaces

Nicole Mestrano

Annales de l'institut Fourier (1985)

  • Volume: 35, Issue: 2, page 217-249
  • ISSN: 0373-0956

Abstract

top
Let X be a smooth projective surface, K the canonical divisor, H a very ample divisor and M H ( c 1 , c 2 ) the moduli space of rank-two vector bundles, H -stable with Chern classes c 1 and c 2 . We prove that, if there exists c 1 ' such that c 1 is numerically equivalent to 2 c 1 ' and if c 2 - 1 4 c 1 2 is even, greater or equal to H 2 + H K + 4 , then there is no Poincaré bundle on M H ( c 1 , c 2 ) × X . Conversely, if there exists c 1 ' such that the number c 1 ' · c 1 is odd or if 1 2 c 1 2 - 1 2 c 1 · K - c 2 is odd, then there exists a Poincaré bundle on M H ( c 1 , c 2 ) × X .

How to cite

top

Mestrano, Nicole. "Poincaré bundles for projective surfaces." Annales de l'institut Fourier 35.2 (1985): 217-249. <http://eudml.org/doc/74676>.

@article{Mestrano1985,
abstract = {Let $X$ be a smooth projective surface, $K$ the canonical divisor, $H$ a very ample divisor and $M_ H(c_ 1,c_ 2)$ the moduli space of rank-two vector bundles, $H$-stable with Chern classes $c_ 1$ and $c_ 2$. We prove that, if there exists $c^\{\prime \}_ 1$ such that $c_ 1$ is numerically equivalent to $2c^\{\prime \}_ 1$ and if $c_ 2-\{1\over 4\}c^ 2_ 1$ is even, greater or equal to $H^ 2+HK+4$, then there is no Poincaré bundle on $M_ H(c_ 1,c_ 2)\times X$. Conversely, if there exists $c^\{\prime \}_ 1$ such that the number $c^\{\prime \}_ 1\cdot c_ 1$ is odd or if $\{1\over 2\}c^ 2_ 1-\{1\over 2\}c_ 1\cdot K-c_ 2$ is odd, then there exists a Poincaré bundle on $M_ H(c_ 1,c_ 2)\times X$.},
author = {Mestrano, Nicole},
journal = {Annales de l'institut Fourier},
keywords = {smooth projective surface; very ample divisor; moduli space of rank-two vector bundles; Chern classes; no Poincaré bundle},
language = {eng},
number = {2},
pages = {217-249},
publisher = {Association des Annales de l'Institut Fourier},
title = {Poincaré bundles for projective surfaces},
url = {http://eudml.org/doc/74676},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Mestrano, Nicole
TI - Poincaré bundles for projective surfaces
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 2
SP - 217
EP - 249
AB - Let $X$ be a smooth projective surface, $K$ the canonical divisor, $H$ a very ample divisor and $M_ H(c_ 1,c_ 2)$ the moduli space of rank-two vector bundles, $H$-stable with Chern classes $c_ 1$ and $c_ 2$. We prove that, if there exists $c^{\prime }_ 1$ such that $c_ 1$ is numerically equivalent to $2c^{\prime }_ 1$ and if $c_ 2-{1\over 4}c^ 2_ 1$ is even, greater or equal to $H^ 2+HK+4$, then there is no Poincaré bundle on $M_ H(c_ 1,c_ 2)\times X$. Conversely, if there exists $c^{\prime }_ 1$ such that the number $c^{\prime }_ 1\cdot c_ 1$ is odd or if ${1\over 2}c^ 2_ 1-{1\over 2}c_ 1\cdot K-c_ 2$ is odd, then there exists a Poincaré bundle on $M_ H(c_ 1,c_ 2)\times X$.
LA - eng
KW - smooth projective surface; very ample divisor; moduli space of rank-two vector bundles; Chern classes; no Poincaré bundle
UR - http://eudml.org/doc/74676
ER -

References

top
  1. [1] W. BARTH, Some properties of stable rank-2 vector bundles on Pn, Math. Ann., 226 (1977). Zbl0332.32021MR55 #2905
  2. [2] G. ELLINGSRUD, S. A. STRØMME, On the moduli space for stable rank-2 vector bundles on P2 (preprint Oslo). Zbl0632.14013
  3. [3] R. HARSTHØRNE, Algebraic geometry, Springer, 1977. Zbl0367.14001
  4. [4] R. HARTSHORNE, Stable reflexive sheaves, Math. Ann., 254 (1980), 121-176. Zbl0431.14004MR82b:14011
  5. [5] A. HIRSCHOWITZ, M. S. NARASIMHAN, Fibrés de 't Hooft spéciaux et applications, Enumerative geometry and classical algebraic geometry, Progress in Math. (1982). Zbl0503.14006
  6. [6] J. LE POTIER, Fibrés stables de rang deux sur P2(C), Math. Ann., 241 (1979), 217-256. Zbl0405.14008MR80m:14012
  7. [7] M. MARUYAMA, On a family of algebraic vector bundles, Number theory, Alg. Geo. and Com. Alg., Tokyo (1973). Zbl0282.14002MR50 #13035
  8. [8] M. MARUYAMA, Moduli of stable sheaves II, J. of Math. of Kyoto Univ., Vol. 18 (1978). Zbl0395.14006MR82h:14011
  9. [9] M. MARUYAMA, Elementary transformation in the theory of vector bundles, Lecture Notes, 961, Alg. Geo., Springer Verlag, 1981. Zbl0505.14009
  10. [10] N. MESTRANO, Sections rationnelles de morphismes algébriques (preprint, Nice), 1983. 
  11. [11] N. MESTRANO, S. RAMANAN, Poincaré bundles for families of curves (preprint, 1984). Zbl0566.14013
  12. [12] M. S. NARASIMHAN, S. RAMANAN, Vector bundles on curves in algebraic geometry, Bombay Colloquium, 1968. Zbl0213.23002
  13. [13] P. E. NEWSTEAD, A non existence theorem for families of stable bundles, Jour. Lond. Math. Soc. Zbl0248.14007
  14. [14] S. RAMANAN, The moduli space of vector bundles over an algebraic curve, Math. Ann., 200 (1973). Zbl0239.14013MR48 #3962
  15. [15] R. L. E. SCHWARZENBERGER, Vector bundles on algebraic surfaces, Proc. London Math. Soc., Vol. 11 (1961). Zbl0212.26003MR25 #1160
  16. [16] F. TAKEMOTO, Stable vector bundles on algebraic surfaces, Nagoya Math. Jour., 47 (1972). Zbl0245.14007MR49 #2735

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.