Recovering plane curves from their bitangents
Authors:
Lucia Caporaso and Edoardo Sernesi
Journal:
J. Algebraic Geom. 12 (2003), 225-244
DOI:
https://doi.org/10.1090/S1056-3911-02-00307-7
Published electronically:
October 17, 2002
MathSciNet review:
1949642
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Abstract |
References |
Additional Information
Abstract: We prove that a general complex projective plane quartic curve is uniquely determined by its 28 bitangent lines. A similar property (called theta-property in the paper) is proved for a general singular quartic having $\delta =1,\dots ,4$ double points with respect to its set of generalized bitangents (suitably defined). The proofs are by degeneration.
[AF1]AF1 Aluffi, P. - Faber, C.: Plane curves with small linear orbits II. International J. of Math. 11 (2000), 591–608.
[AF2]AF2 Aluffi, P. - Faber, C.: Linear orbits of arbitrary plane curves. Mich. Math. J. (W.Fulton issue) 48 (2000), 1–37.
[ACGH]ACGH Arbarello - Cornalba - Griffiths - Harris: Geometry of Algebraic Curves, I. Grund. series, vol. 267, Springer.
[E-Ch]E-Ch Enriques, F. - Chisini, O.: Teoria geometrica delle equazioni e delle funzioni algebriche, vol. I. Zanichelli, Bologna 1929.
[H]H Harris, J.: Theta-characteristics on algebraic curves. Trans. AMS 271 (1982), pp. 611–638.
[K-W]K-W Krazer, A. - Wirtinger, W.: Abelsche Funktionen und Allgemeine Thetafunktionen. Enc. d. Math. Wiss. II B 7, Leipzig 1921.
[GIT]GIT Mumford, D. - Fogarty, J. - Kirwan, F.: Geometric Invariant Theory. (Third edition) E.M.G. 34 Springer 1994.
[S]S Segre, C.: Le molteplicità nelle intersezioni delle curve piane algebriche con alcune applicazioni ai principii della teoria di tali curve. Giornale di Matematiche 36 (1898), 1-50. Opere, vol. I, 380-429.
[W]W Walker, R.J.: Algebraic Curves. Princeton U.P. 1950.
[Z]Z Zariski, O.: Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order $n$ and with a given number $d$ of nodes. Am. J. Math. 104 (1982), pp. 209-226.
[AF1]AF1 Aluffi, P. - Faber, C.: Plane curves with small linear orbits II. International J. of Math. 11 (2000), 591–608.
[AF2]AF2 Aluffi, P. - Faber, C.: Linear orbits of arbitrary plane curves. Mich. Math. J. (W.Fulton issue) 48 (2000), 1–37.
[ACGH]ACGH Arbarello - Cornalba - Griffiths - Harris: Geometry of Algebraic Curves, I. Grund. series, vol. 267, Springer.
[E-Ch]E-Ch Enriques, F. - Chisini, O.: Teoria geometrica delle equazioni e delle funzioni algebriche, vol. I. Zanichelli, Bologna 1929.
[H]H Harris, J.: Theta-characteristics on algebraic curves. Trans. AMS 271 (1982), pp. 611–638.
[K-W]K-W Krazer, A. - Wirtinger, W.: Abelsche Funktionen und Allgemeine Thetafunktionen. Enc. d. Math. Wiss. II B 7, Leipzig 1921.
[GIT]GIT Mumford, D. - Fogarty, J. - Kirwan, F.: Geometric Invariant Theory. (Third edition) E.M.G. 34 Springer 1994.
[S]S Segre, C.: Le molteplicità nelle intersezioni delle curve piane algebriche con alcune applicazioni ai principii della teoria di tali curve. Giornale di Matematiche 36 (1898), 1-50. Opere, vol. I, 380-429.
[W]W Walker, R.J.: Algebraic Curves. Princeton U.P. 1950.
[Z]Z Zariski, O.: Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order $n$ and with a given number $d$ of nodes. Am. J. Math. 104 (1982), pp. 209-226.
Additional Information
Lucia Caporaso
Affiliation:
Università degli Studi del Sannio, Benevento, Italy;
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication:
Dipartimento di Matematica, Università Roma Tre, L.Go S.L. Murialdo 1, 00146 Roma, Italy
MR Author ID:
345125
Email:
caporaso@math.mit.edu, caporaso@matrm3.mat.uniroma3.it
Edoardo Sernesi
Affiliation:
Dipartimento di Matematica, Università Roma Tre, L.Go S.L. Murialdo 1, 00146 Roma, Italy
MR Author ID:
158910
Email:
sernesi@matrm3.mat.uniroma3.it
Received by editor(s):
September 15, 2000
Published electronically:
October 17, 2002
