The wikipedia page on plane quartics (http://en.m.wikipedia.org/wiki/Quartic_plane_curve) mentions the possible number of singularities that such a curve can have, including some examples. I'd like to have an explicit example of a quartic having precisely two ordinary double points, or, if possible, a parametrization of an entire family of such curves. Any explicit examples or references would be appreciated.

Background: I am interested in elliptic curves occuring as desingularizations, and such quartics should yield examples by the genus formula. (I started by considering singular Weierstrass equations, but their normalizations are rational)

  • $\begingroup$ "A Guide to Plane Algebraic Curves" by Keith Kendig discusses the topic of intersection multiplicities in detail. $\endgroup$
    – John McGee
    Feb 20 '15 at 17:25

This may be unsatisfying, but unless I have done something wrong, $x^2 yz -xyz^2+x^4-2x^3z+x^2z^2+y^4$ gives one such example. My method was simply to force singularities at (0,0) and (1,0), which I did with all but the last term (expand out in terms of (x-1) to see why I did what I did), and then the last term was there to get rid of other singularities. Substituting $y^4$ with a general quartic term that doesn't mess up the singularities (e.g., $ky^4$ or a $y^3z$ term) should give you a family "most" of whose elements have two ordinary double points.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.