Plane curves as Pfaffians
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 363-388.

Let C be a smooth curve in 2 given by an equation F=0 of degree d. In this paper we parametrise all linear Pfaffian representations of F by an open subset in the moduli space M C (2,K C ). We construct an explicit correspondence between Pfaffian representations of C and rank 2 vector bundles on C with canonical determinant and no sections.

Publié le :
Classification : 14H60, 14D20, 15A15, 15A54
Buckley, Anita 1 ; Košir, Tomaž 1

1 Department of Mathematics Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 1000 Ljubljana, Slovenia
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Buckley, Anita; Košir, Tomaž. Plane curves as Pfaffians. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 363-388. http://www.numdam.org/item/ASNSP_2011_5_10_2_363_0/

[1] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, “The Geometry of Algebraic Curves”, Vol. 1, Grundlehren der mathematischen Wissenschaften, Vol. 267, Springer-Verlag, 1985. | MR | Zbl

[2] A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–63. | MR | Zbl

[3] A. Beauville, Vector bundles on curves and generalized theta functions: recent results and open problems, In: “Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93)”, Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge Univ. Press, Cambridge, 1995, 17–33. | MR | Zbl

[4] A. Buckley, Elementary transformations of Pfaffian representations of plane curves, Linear Algebra Appl. 433 (2010), 758–780. | MR | Zbl

[5] R. J. Cook and A. D. Thomas, Line bundles and homogeneous matrices, Quart. J. Math. Oxford Ser. (2) 30 (1979), 423–429. | MR | Zbl

[6] I. Dolgachev, “Topics in Classical Algebraic Geometry”, Lecture Notes http://www.math.lsa.umich.edu/ idolga/lecturenotes.html. | MR

[7] J. M. Drezet and M. S. Narasimhan, Groupe de Picard des varietes de modules de fibres semi-stables sur les courbes algebriques, Invent. Math. 97 (1989), 53–94. | EuDML | MR | Zbl

[8] W. Fulton and P. Pragacz, “Shubert Varieties and Degeneracy Loci”, Lecture Notes in Mathematics, Vol. 1689, Springer-Verlag, 1998. | MR | Zbl

[9] R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, 1977. | MR | Zbl

[10] M. Marcus and R. Westwick, Linear maps on skew-symmetric matrices: The invariance of elementary symmetric functions, Pacific J. Math. (3) 14 (1960). | MR | Zbl

[11] P. Lancaster and L. Rodman, Canonical forms for symmetric / skew-symmetric real matrix pairs under strict equivalence and congruence, Linear Algebra Appl. 406 (2005), 1–76. | MR | Zbl

[12] Y. Laszlo, À propos de l’espace des modules des fibrés de rang 2 sur une courbe, Math. Ann. 299 (1994), 597–608. | EuDML | MR | Zbl

[13] R. Lazarsfeld, “Positivity in Algebraic Geometry II”, A Series of Modern Surveys in Mathematics, Vol. 49, Springer-Verlag, 2000. | MR | Zbl

[14] M. S. Narasimhan and S. Ramanan, 2θ linear systems on Abelian varieties, In: “Vector bundles on algebraic varieties (Bombay, 1984)”, Stud. Math., Vol. 11, Tata Inst. Fund. Res., Bombay, 1987, 415–427. | MR | Zbl

[15] P. E. Newstead, “Introduction to Moduli Problems and Orbit Spaces”, Tata Institute of Fundamental Research, Bombay, Springer-Verlag, 1978. | MR | Zbl

[16] W. M. Oxbury, C. Pauly and E. Previato, Subvarieties of 𝒮𝒰 C (2) and 2θ-divisors in the Jacobian, Trans. Amer. Math. Soc. 350 (1998), 3587–3614. | MR | Zbl

[17] J. Le Potier, “Lectures on Vector Bundles”, Cambridge studies in advanced mathematics, Vol. 54, Cambridge University Press, 1997. | MR | Zbl

[18] C. S. Seshadri, “Fibrés vectoriels sur les courbes algébriques”, Astérisque, Vol. 96, 1982. | Numdam | MR | Zbl

[19] I. R. Shafarevich, “Basic Algebraic Geometry 1”, Springer-Verlag, 1994. | MR | Zbl

[20] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992), 131–149. | MR | Zbl

[21] P. Vanhaecke, Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve, Ann. Inst. Fourier (Grenoble) 6 (2005), 1789–1802. | EuDML | Numdam | MR | Zbl

[22] V. Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103–140. | MR | Zbl

[23] V. Vinnikov, “Determinantal Representations of Real Cubics and Canonical Forms of Corresponding Triples of Matrices”, Mathematical Theory of Networks and Systems, Lecture Notes in Control and Inform. Sci., Vol. 58, 1984. | MR | Zbl