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Infinitesimal Invariant and vector bundles

Published online by Cambridge University Press:  11 January 2016

Gian Pietro Pirola  and
Cecilia Rizzi
Gian Pietro Pirola
Affiliation:
Dipartimento di Matematica F. Casorati Università di Pavia via Ferrata 1, 27100 Pavia Italy gianpietro.pirola@unipv.it
Cecilia Rizzi
Affiliation:
Dipartimento di Matematica F. Brioschi Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano Italy cecilia.rizzi@polimi.it
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Abstract

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We study the Saito-Ikeda infinitesimal invariant of the cycle defined by curves in their Jacobians using rank k + 1 vector bundles. We give a criterion for which the higher cycle class map is not trivial. When k = 2, this turns out to be strictly linked to the Petri map for vector bundles. In this case we can improve a result of Ikeda: an explicit construction on a curve of genus g ≥ 10 shows the existence of a non trivial element in the higher Griffiths group.

Keywords

14C2514H4014C15

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 186 , 2007 , pp. 95 - 118
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J., Geometry of Algebraic Curves, New York, Springer-Verlag, 1985.Google Scholar
[2] Beauville, A., Sur l’anneau de Chow d’une variete abelianne, Math. Ann., 273 (1986), 647–651.Google Scholar
[3] Beilinson, A., Height pairing between algebraic cycles, Lecture Notes in Math., 1289 (1987), 1–26.CrossRefGoogle Scholar
[4] Bertram, A. and Feinberg, B., On stable rank two bundles with canonical determinant and many sections, Lecture Notes in Pure and Appl. Math., 200 (1998), 259–269.Google Scholar
[5] Ceresa, G., C is not algebraically equivalent to C- , Ann. of Math., 117 (1983), 285–291.Google Scholar
[6] Collino, A. and Pirola, G., The Griffiths infinitesimal invariant for curve in its Jacobian, Duke Math. J., 78 (1995), 59–88.CrossRefGoogle Scholar
[7] Colombo, E., On curves with a theta-characteristic whose space of sections has dimension 4, Math. Z., 215 (1994), no. 4, 655–665.CrossRefGoogle Scholar
[8] Colombo, E. and Geemen, B. Van, Note on curves in a jacobian, Comp. Math., 88 (1993), 333–353.Google Scholar
[9] Harris, B., Harmonic volumes, Acta Math., 150 (1983), 91–123.Google Scholar
[10] Hartshorne, R., Algebraic geometry, Graduate texts in mathematics 52, New York, Springer-Verlag, 1977.Google Scholar
[11] A. Ikeda, Algebraic cycles and infinitesimal invariant on jacobian varieties, J. Algebraic Geom., 12 (2003), 573–603.CrossRefGoogle Scholar
[12] Fakhruddin, N., Algebraic cycles on generic Abelian varieties, Comp. Math., 100 (1996), 101–119.Google Scholar
[13] Gieseker, D., On the moduli of vector bundles on an algebraic surface, Ann. of Math., 106 (1977), 45–60.CrossRefGoogle Scholar
[14] Green, M., Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom., 29 (1989), 545–555.CrossRefGoogle Scholar
[15] Griffiths, P., Infinitesimal variation of Hodge structures (III): determinantal varieties and the infinitesimal invariant of normal functions, Comp. Math., 50 (1983), 267–324.Google Scholar
[16] Mukai, S., Vector bundles and Brill-Noether theory, Math. Sci. Res. Inst. Publ., 28 (1995), 145–158.Google Scholar
[17] Murre, J., On a conjectural filtration on the Chow groups of an algebraic variety, Indag. Math. (N.S.), 4 (1993), 177–201.CrossRefGoogle Scholar
[18] Saito, S., Motives and filtrations on Chow Groups, Invent. Math., 125 (1996), 149–196.CrossRefGoogle Scholar
[19] Saito, S., Higher normal functions and Griffiths groups, J. Algebraic Geom., 11 (2002), 161–201.CrossRefGoogle Scholar
[20] Teixidor, M., Brill-Noether theory for stable vector bundles, Duke Math. J., 62 (1991), no. 2, 385–400.Google Scholar
[21] Voisin, C., Une approche infinitesimale du theorem de H. Clemens sur les cycles d’une quintique generale de P4 , J. Algebraic Geom., 1 (1992), 157–174.Google Scholar
[22] Voisin, C., Une remarque sur l’invariant infinitesimal des functions normal, C. R. Acad. Sci. Paris Ser. I, 307 (1988), 157–160.Google Scholar
[23] Weibel, C., An introduction to homological algebra, Cambridge studies in Advanced Math. 38, Cambridge University press, Cambridge, 1994.Google Scholar