Plane curves as Pfaffians
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 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.

Published online : 2018-08-07
Classification:  14H60,  14D20,  15A15,  15A54
@article{ASNSP_2011_5_10_2_363_0,
     author = {Buckley, Anita and Ko\v sir, Toma\v z},
     title = {Plane curves as Pfaffians},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {363-388},
     zbl = {1237.14039},
     mrnumber = {2856152},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_363_0}
}
Buckley, Anita; Košir, Tomaž. Plane curves as Pfaffians. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 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 770932 | Zbl 0559.14017

[2] A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–63. | MR 1786479 | Zbl 1076.14534

[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 1397056 | Zbl 0846.14024

[4] A. Buckley, Elementary transformations of Pfaffian representations of plane curves, Linear Algebra Appl. 433 (2010), 758–780. | MR 2654105 | Zbl 1191.14042

[5] R. J. Cook and A. D. Thomas, Line bundles and homogeneous matrices, Quart. J. Math. Oxford Ser. (2) 30 (1979), 423–429. | MR 559048 | Zbl 0437.14004

[6] I. Dolgachev, “Topics in Classical Algebraic Geometry”, Lecture Notes http://www.math.lsa.umich.edu/ idolga/lecturenotes.html. | MR 723704

[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. | MR 999313 | Zbl 0689.14012

[8] W. Fulton and P. Pragacz, “Shubert Varieties and Degeneracy Loci”, Lecture Notes in Mathematics, Vol. 1689, Springer-Verlag, 1998. | MR 1639468 | Zbl 0913.14016

[9] R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, 1977. | MR 463157 | Zbl 0367.14001

[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 114823 | Zbl 0093.24204

[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 2156428 | Zbl 1081.15007

[12] Y. Laszlo, À propos de l’espace des modules des fibrés de rang 2 sur une courbe, Math. Ann. 299 (1994), 597–608. | MR 1286886 | Zbl 0846.14011

[13] R. Lazarsfeld, “Positivity in Algebraic Geometry II”, A Series of Modern Surveys in Mathematics, Vol. 49, Springer-Verlag, 2000. | MR 2095472 | Zbl 1093.14500

[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 893605 | Zbl 0685.14023

[15] P. E. Newstead, “Introduction to Moduli Problems and Orbit Spaces”, Tata Institute of Fundamental Research, Bombay, Springer-Verlag, 1978. | MR 546290 | Zbl 1277.14001

[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 1467474 | Zbl 0898.14014

[17] J. Le Potier, “Lectures on Vector Bundles”, Cambridge studies in advanced mathematics, Vol. 54, Cambridge University Press, 1997. | MR 1428426 | Zbl 0872.14003

[18] C. S. Seshadri, “Fibrés vectoriels sur les courbes algébriques”, Astérisque, Vol. 96, 1982. | Numdam | MR 699278 | Zbl 0517.14008

[19] I. R. Shafarevich, “Basic Algebraic Geometry 1”, Springer-Verlag, 1994. | MR 1328833 | Zbl 0797.14001

[20] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992), 131–149. | MR 1152228 | Zbl 0772.53013

[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. | Numdam | MR 2187935 | Zbl 1087.14027

[22] V. Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103–140. | MR 1024486 | Zbl 0704.14041

[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 792158 | Zbl 0577.14040