The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic

Laurent Ducrohet[1]

  • [1] École Polytechnique CMLS 91128 Palaiseau Cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 4, page 1641-1669
  • ISSN: 0373-0956

Abstract

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Let X be a general proper and smooth curve of genus 2 (resp. of genus 3 ) defined over an algebraically closed field of characteristic p . When 3 ≤ p ≤ 7 , the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X . Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian surfaces) that appear as the Prym varieties associated to double étale coverings of X . We are therefore able to compute the explicit equations defining Frobenius action in these cases. We perform some of these computations and draw some geometric consequences.

How to cite

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Ducrohet, Laurent. "The Frobenius action on rank $2$ vector bundles over curves in small genus and small characteristic." Annales de l’institut Fourier 59.4 (2009): 1641-1669. <http://eudml.org/doc/10437>.

@article{Ducrohet2009,
abstract = {Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p \le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian surfaces) that appear as the Prym varieties associated to double étale coverings of $X$. We are therefore able to compute the explicit equations defining Frobenius action in these cases. We perform some of these computations and draw some geometric consequences.},
affiliation = {École Polytechnique CMLS 91128 Palaiseau Cedex (France)},
author = {Ducrohet, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Vector bundles; Frobenius; Prym varieties; vector bundles},
language = {eng},
number = {4},
pages = {1641-1669},
publisher = {Association des Annales de l’institut Fourier},
title = {The Frobenius action on rank $2$ vector bundles over curves in small genus and small characteristic},
url = {http://eudml.org/doc/10437},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Ducrohet, Laurent
TI - The Frobenius action on rank $2$ vector bundles over curves in small genus and small characteristic
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 4
SP - 1641
EP - 1669
AB - Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p \le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian surfaces) that appear as the Prym varieties associated to double étale coverings of $X$. We are therefore able to compute the explicit equations defining Frobenius action in these cases. We perform some of these computations and draw some geometric consequences.
LA - eng
KW - Vector bundles; Frobenius; Prym varieties; vector bundles
UR - http://eudml.org/doc/10437
ER -

References

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