Abstract
A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question of whether or not a smooth plane curve of degree at least 4 is determined fromits inflection lines is still open. Over a field of characteristic 0,we showthat it is possible to reconstruct any smooth plane quartic with at least 8 hyperinflection points by its inflection lines. Our methods apply also in positive characteristic, where we show a similar result, with two exceptions in characteristic 13.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Abreu, M. Pacini and D. Testa. The general plane quartic is determined by its inflection lines. In preparation.
F. Bogomolov, M. Korotiaev and Y. Tschinkel. A Torelli theorem for curves over finite fields. Pure Appl. Math. Q., 6(1) (2010), Special Issue: In honor of John Tate. Part 2, 245–294.
J.W. Bruce and P.J. Giblin. A stratification of the space of plane quartic curves. Proc. London Math. Soc. (3), 42(2) (1981), 270–298.
L. Caporaso and E. Sernesi. Recovering plane curves from their bitangents. J. Algebraic Geom., 12(2) (2003), 225–244.
I.V. Dolgachev. Classical algebraic geometry. Cambridge University Press, Cambridge, 2012. A modern view.
W.L. Edge. A plane quartic curve with twelve undulations. Edinburgh Math. Notes 1945 (1945), no. 35, 10–13.
W.L. Edge. A plane quartic with eight undulations. Proc. Edinburgh Math. Soc. (2), 8 (1950), 147–162.
M. Girard. The group of Weierstrass points of a plane quartic with at least eight hyperflexes. Math. Comp., 75(255) (2006), 1561–1583 (electronic).
A. Hefez and S.L. Kleiman. Notes on the duality of projective varieties. Geometry today (Rome, 1984), Progr. Math., Birkhäuser Boston, Boston, MA, 60 (1985), 143–183.
A. Kuribayashi and K. Komiya. On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three. Hiroshima Math. J., 7(3) (1977), 743–768.
D. Lehavi. Any smooth plane quartic can be reconstructed from its bitangents. Israel J. Math., 146 (2005), 371–379.
M. Pacini and D. Testa. Recovering plane curves of low degree from their inflection lines and inflection points. Israel J. Math., 195(1) (2013), 283–316.
A.M. Vermeulen. Weierstrass points of weight two on curves of genus three. Universiteit van Amsterdam, Amsterdam, 1983. Dissertation, University of Amsterdam, Amsterdam, 1983;With a Dutch summary.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by CNPq, process no. 300714/2010-6.
The second author was partially supported by the EPSRC grant EP/K019279/1 “Moduli spaces and rational points”.
About this article
Cite this article
Pacini, M., Testa, D. Plane quartics with at least 8 hyperinflection points. Bull Braz Math Soc, New Series 45, 819–836 (2014). https://doi.org/10.1007/s00574-014-0077-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-014-0077-3