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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)

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  • $\begingroup$ "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
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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.

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