A criterion for virtual global generation

Indranil Biswas; A. J. Parameswaran

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 1, page 39-53
  • ISSN: 0391-173X

Abstract

top
Let X be a smooth projective curve defined over an algebraically closed field k , and let F X denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually globally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if ( F X m ) * E ≅ E a ⊕ E f for some m , where E a is some ample vector bundle and E f is some finite vector bundle over X (either of E a and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

How to cite

top

Biswas, Indranil, and Parameswaran, A. J.. "A criterion for virtual global generation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.1 (2006): 39-53. <http://eudml.org/doc/240363>.

@article{Biswas2006,
abstract = {Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $F_X$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only if $(F^m_X)^* E\, \cong \, E_a\oplus E_f$ for some $m$, where $E_a$ is some ample vector bundle and $E_f$ is some finite vector bundle over $X$ (either of $E_a$ and $E_f$ are allowed to be zero). If the characteristic of $k$ is zero, a vector bundle $E$ over $X$ is virtually globally generated if and only if $E$ is a direct sum of an ample vector bundle and a finite vector bundle over $X$ (either of them are allowed to be zero).},
author = {Biswas, Indranil, Parameswaran, A. J.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {39-53},
publisher = {Scuola Normale Superiore, Pisa},
title = {A criterion for virtual global generation},
url = {http://eudml.org/doc/240363},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Biswas, Indranil
AU - Parameswaran, A. J.
TI - A criterion for virtual global generation
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 1
SP - 39
EP - 53
AB - Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $F_X$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only if $(F^m_X)^* E\, \cong \, E_a\oplus E_f$ for some $m$, where $E_a$ is some ample vector bundle and $E_f$ is some finite vector bundle over $X$ (either of $E_a$ and $E_f$ are allowed to be zero). If the characteristic of $k$ is zero, a vector bundle $E$ over $X$ is virtually globally generated if and only if $E$ is a direct sum of an ample vector bundle and a finite vector bundle over $X$ (either of them are allowed to be zero).
LA - eng
UR - http://eudml.org/doc/240363
ER -

References

top
  1. [Ba] C. M. Barton, Tensor products of ample vector bundles in characteristic p , Amer. J. Math. 93 (1971), 429–438. Zbl0221.14011MR289525
  2. [Bi1] I. Biswas, Parabolic ample bundles, Math. Ann. 307 (1997), 511–529. Zbl0877.14013MR1437053
  3. [Bi2] I. Biswas, A criterion for ample vector bundles over a curve in positive characteristic, Bull. Sci. Math. 129 (2005), 539–543. Zbl1067.14027MR2142897
  4. [BP] I. Biswas and A. J. Parameswaran, On the ample vector bundles over curves in positive characteristic, C.R. Acad. Sci. Paris Sér. I Math. 339 (2004), 355–358. Zbl1062.14042MR2092463
  5. [Br] H. Brenner, Slopes of vector bundles on projective curves and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (2004), 371–392. Zbl1041.13002MR2020037
  6. [Ha1] R. Hartshorne, “Ample Subvarieties of Algebraic Varieties”, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Zbl0208.48901MR282977
  7. [Ha2] R. Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73–89. Zbl0218.14018MR292847
  8. [La] H. Lange, On stable and ample vector bundles of rank 2 on curves, Math. Ann. 238 (1978), 193–202. Zbl0377.14005MR514426
  9. [Lan] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276. Zbl1080.14014MR2051393
  10. [MS] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205–239. Zbl0454.14006MR575939
  11. [Mu] D. Mumford, Pathologies III, Amer. J. Math. 89 (1967), 94–104. Zbl0146.42403MR217091
  12. [No] M. V. Nori, On the representations of the fundamental group scheme, Compositio Math. 33 (1976), 29–41. Zbl0337.14016MR417179
  13. [Pa] A. J. Parameswaran, Virtual global generation of ample bundles over curves, unpublished manuscript. 
  14. [RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tôhoku Math. J. (2) 36 (1984), 269–291. Zbl0567.14027MR742599

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.