Parametric Equations of Plane Sextic Curves with a Maximal Set of Singularities

S. Yu. Orevkov

Last update: July 7, 2014


Abstract
We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.


abh-2026-07-03.pdf - The Paper (published Abhyankar Memorial Special Issue of the "Journal of Algebra and its Applications", vol.14 (2025), no.9, 1540013)


sextics-param.txt - The parametric equations in Maple format

sextics-param-check.mws - An mws file that checks the parametric equations


The following mws files are created using Maple 9 (all the three mws files zipped are here: sextics-param-mws.zip)

a7a6a6.mws - Computations for the proof that the sextic curve with singularities A7+2A6 (no. 34) does not have a parametrization with coefficients in Q(sqrt(-7)); see Sect. 2 of the paper.

a7a4a4a2a2.mws - Computations for the proof that the sextic curve with singularities A7+2A4+2A2 (no. 36) does not have a parametrization with rational coefficients; see Sect. 2 of the paper.

a10a4a4a1.mws - Computation of the parametrization of the sextic curve with singularities A10+2A4+A1 (no. 24); see Sect. 3.3 of the paper.



homepage