Some Remarks on Prym-Tyurin Varieties

Giuliano Parigi

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 1055-1069
  • ISSN: 0392-4033

Abstract

top
The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism u a Jacobian J ( C ) has connected kernel, the principal polarization on J ( C ) induces a multiple of the principal polarization on the image of u . We reformulate and complete this theorem proving "constructively" the following: Theorem. Let Z J ( C ) be an abelian subvariety and Y its complementary variety. Z is a Prym-Tyurin variety with respect to J ( C ) if and only if the following sequence 0 Y J ( C ) Z 0 is exact. b) In [5] Izadi set the question whether every p.p.a.v. is a Prym-Tyurin variety for a symmetric fixed point free correspondence. In this work a contribution to a possible negative answer to this question is provided by building a classical Prym-Tyurin variety explicitly, but this variety can never be defined through a fixed point free correspondence.

How to cite

top

Parigi, Giuliano. "Some Remarks on Prym-Tyurin Varieties." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 1055-1069. <http://eudml.org/doc/290431>.

@article{Parigi2007,
abstract = {The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$. We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z \subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence $0 \to Y \hookrightarrow J(C) \to Z \to 0$ is exact. b) In [5] Izadi set the question whether every p.p.a.v. is a Prym-Tyurin variety for a symmetric fixed point free correspondence. In this work a contribution to a possible negative answer to this question is provided by building a classical Prym-Tyurin variety explicitly, but this variety can never be defined through a fixed point free correspondence.},
author = {Parigi, Giuliano},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {1055-1069},
publisher = {Unione Matematica Italiana},
title = {Some Remarks on Prym-Tyurin Varieties},
url = {http://eudml.org/doc/290431},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Parigi, Giuliano
TI - Some Remarks on Prym-Tyurin Varieties
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 1055
EP - 1069
AB - The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$. We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z \subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence $0 \to Y \hookrightarrow J(C) \to Z \to 0$ is exact. b) In [5] Izadi set the question whether every p.p.a.v. is a Prym-Tyurin variety for a symmetric fixed point free correspondence. In this work a contribution to a possible negative answer to this question is provided by building a classical Prym-Tyurin variety explicitly, but this variety can never be defined through a fixed point free correspondence.
LA - eng
UR - http://eudml.org/doc/290431
ER -

References

top
  1. ALBERT, A. A., A note on the Poincaré theorem on impure matrices, Annals of Mathematics, 36, no. 1 (1935), 151-156. Zbl0011.00603MR1503214DOI10.2307/1968670
  2. BEAUVILLE, A., Prym varieties: a survey, Proceedings of Symposium Pure Mathematics, 49, part. 1 (1989). Zbl0736.14020MR1013156
  3. BLOCH, S. - MURRE, J. P., On the Chow group of certain types of Fano three-folds, Comp. Math., 39 (1979), 47-105. Zbl0426.14018MR539001
  4. HURWITZ, A., Über algebraische korrespondenzen und das verallgemeinerte korrespondenzprinzip, Math. Ann., 28, (1886), 561- 585. MR1510394DOI10.1007/BF01447915
  5. IZADI, E., Subvarieties of abelian varieties, in "Applications of Algebraic Geometry to Coding Theory, Physics and Computation", C. Ciliberto et al. (eds.), 2001, 207-214. Zbl1006.14015MR1866901
  6. KANEV, V., Principal polarization of Prym-Tyurin varieties, Comp. Math., 64 (1987), 243-270. Zbl0694.14009MR918413
  7. LANG, S., "Algebra", revised third edition, Graduate texts in Mathematics, Springer-Verlag, 2002. Zbl0984.00001MR1878556DOI10.1007/978-1-4613-0041-0
  8. LANGE, H. - BIRKENHAKE, CH., Complex Abelian Varieties, Grundleheren der mathematischen Wissenchaffen302, Springer-Verlag, 1992. MR1217487DOI10.1007/978-3-662-02788-2
  9. SCORZA, G., Intorno alla teoria generale delle matrici di Riemann ed alcune sue applicazioni, Rendiconti del Circolo Matematico di Palermo, XLI (1916), 263-380. 
  10. TYURIN, A., Five lectures on three-dimensional, Russian Math. Surveys, 27 (1972), 1- 53. MR412196
  11. WELTERS, G., Curves of twice the minimal class on principally polarized abelian varieties, Indag. Math., 94 (1987), 87-109. Zbl0644.14014MR883371

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.