The $2$-Hessian and sextactic points on plane algebraic curves

  • Paul Aleksander Maugesten
  • Torgunn Karoline Moe

Abstract

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

References

Arnold, V. I., Remarks on the extactic points of plane curves, in “The Gelfand Mathematical Seminars, 1993–1995”, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 1996, pp. 11--22.

Ballico, E. and Gatto, L., Weierstrass points on singular curves, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), no. 2, 145–170.

Brieskorn, E. and Knörrer, H., Plane algebraic curves, Birkhäuser Verlag, Basel, 1986. https://doi.org/10.1007/978-3-0348-5097-1

Cayley, A., On the conic of five-pointic contact at any point of a plane curve, Philosophical Transactions of the Royal Society of London 149 (1859), 371–400.

Cayley, A., On the sextactic points of a plane curve, Philosophical Transactions of the Royal Society of London 155 (1865), 545–578.

Cayley, A., The collected mathematical papers, vol. 5, Cambridge University Press, Cambridge, 1892.

Coolidge, J. L., A treatise on algebraic plane curves, Clarendon Press, Oxford, 1931.

Cukierman, F., Determinant of complexes and higher Hessians, Math. Ann. 307 (1997), no. 2, 225–251. https://doi.org/10.1007/s002080050032

Del Centina, A., Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Lückensatz” to the 1970s, Ann. Univ. Ferrara Sez. VII Sci. Mat. 54 (2008), no. 1, 37–59. https://doi.org/10.1007/s11565-008-0037-1

Fischer, G., Plane algebraic curves, Student Mathematical Library, vol. 15, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/stml/015

Flenner, H. and Zaidenberg, M., On a class of rational cuspidal plane curves, Manuscripta Math. 89 (1996), no. 4, 439–459. https://doi.org/10.1007/BF02567528

Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York-Heidelberg, 1977.

IGI, GeoGebra, 2012, Version 4.0, http://www.geogebra.org.

Koras, M., and Palka, K., Complex planar curves homeomorphic to a line have at most four singular points, preprint arXiv:1905.11376 [math.AG], 2019.

Koras, M. and Palka, K., The Coolidge-Nagata conjecture, Duke Math. J. 166 (2017), no. 16, 3085–3145. https://doi.org/10.1215/00127094-2017-0010

Laksov, D., Wronskians and Plücker formulas for linear systems on curves, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 45–66.

Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario., Maple 2016, ©1981–2016, Version 2016.1, http://www.maplesoft.com.

Maugesten, P. A., Sextactic points on plane algebraic curves, Master's thesis, Department of Mathematics, University of Oslo, 2017, http://hdl.handle.net/10852/57792.

Miranda, R., Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. https://doi.org/10.1090/gsm/005

Moe, T. K., Rational cuspidal curves, Master's thesis, Department of Mathematics, University of Oslo, 2008, http://urn.nb.no/URN:NBN:no-19416.

Moe, T. K., Cuspidal curves on Hirzebruch surfaces, Ph.D. thesis, University of Oslo, 2013, http://urn.nb.no/URN:NBN:no-38685, pp. xx+154.

Notari, R., On the computation of Weierstrass gap sequences, Rend. Sem. Mat. Univ. Politec. Torino 57 (1999), no. 1, 23–35.

Orevkov, S. Y., On rational cuspidal curves. I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), no. 4, 657–673. https://doi.org/10.1007/s002080000191

Perkinson, D. M., Jet bundles and curves in Grassmannians, Ph.D. thesis, The University of Chicago, 1990.

Piene, R., Numerical characters of a curve in projective $n$-space, in “Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976)'', Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 475--495.

Salmon, G., A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections”, 3rd ed., Hodges, Foster, and Figgs, Dublin, 1879.

Thorbergsson, G. and Umehara, M., Sextactic points on a simple closed curve, Nagoya Math. J. 167 (2002), 55–94. https://doi.org/10.1017/S0027763000025435

Tono, K., On Orevkov's rational cuspidal plane curves, J. Math. Soc. Japan 64 (2012), no. 2, 365–385.

Widland, C. and Lax, R., Weierstrass points on Gorenstein curves, Pacific J. Math. 142 (1990), no. 1, 197–208.

Published
2019-08-29
How to Cite
Maugesten, P., & Moe, T. (2019). The $2$-Hessian and sextactic points on plane algebraic curves. MATHEMATICA SCANDINAVICA, 125(1), 13-38. https://doi.org/10.7146/math.scand.a-114715
Section
Articles