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I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1.

One can show that for a general set of 14 points in the plane, there is no degree-7 curve with singularities at those points. A construction which gives some intuition about what necessary and/or sufficient conditions the 14 points satisfy would be ideal.

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    $\begingroup$ Take your favourite elliptic curve $E$ and choose a divisor $D$ of degree $7$. It is very ample, so it defines an embedding $\psi \colon E \to \mathbb{P}^6$. By standard results the image $\psi(E)$ can be projected isomorphically in $\mathbb{P}^3$, obtained a smooth curve $X$ of degree $7$ and genus $1$ in $\mathbb{P}^3$. Now, the generic projection in $\mathbb{P}^2$ from an external point should give a $14$-nodal plane curve of degree $7$. I think that doing explicit computations by hand is rather tricky, but it might be possible by using a computer algebra system like Macauley2 or Magma. $\endgroup$
    – Francesco Polizzi
    Commented May 5, 2015 at 16:54
  • $\begingroup$ @FrancescoPolizzi, this is a good answer to my original question, and you are right that I should try this approach. I edited the question to be a bit more specific about the kind of information I am hoping to learn from such a construction. I will leave the post open to see if anyone has a construction which addresses the second part of the question. $\endgroup$
    – stepanp21
    Commented May 5, 2015 at 18:51

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