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