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I'd like to put my hands on some polynomial defining a curve of genus $4$, living in the plane or in the 3D space.

Do you know about any? Is there any procedure to build one?

The best would be one which is:

  • Smooth

  • non hyperelliptic

  • with many real points ( I want to plot it and see it! )

More specifically, it would be great to be able to compute the canonical embedding (or map in the hyperelliptic case) of the curve.

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    $\begingroup$ Take for instance the polynomial $$ f(x, y, z) = \prod_{n = 0}^3 ((x - 2n)^2 + y^2 - 1)^2 + z^2) $$ The set described by $f(x, y, z) = 0$ consists of four circles touching eachother in $\Bbb R^3$. I am pretty certain $f(x, y, z) = \epsilon$ for sufficiently small positive epsilons would describe what you're looking for. $\endgroup$ – Arthur Jan 17 '14 at 10:41
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The degree-genus formula $g=\frac12 (d-1)(d-2)$ for plane curves tells you there is no smooth plane curve of genus 4.

On the other hand, a nonsingular complete intersection of a quadric surface and a cubic surface in $\mathbf{P}^3$ has genus 4, by a straightforward adjunction calculation. In fact adjunction shows that such a curve is canonically embedded, and moreover (Hartshorne Example IV.5.2.2) every nonhyperelliptic genus 4 curve arises as a complete intersection of this kind.

I guess this answer slightly skirts the issue of "putting your hands on" such a curve, but I claim that "random" choices of quadratic and cubic forms should give you what you desire. (The issue with real points might require a slight bit of care, but as long as the quadratic form is indefinite, I think it should be fine.)

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  • $\begingroup$ Thank you very much for your answer! Let's say I pick a 2 polynomials in $X,Y,Z$ describing a quadric and a cubic surfaces, I plot them and the real intersection seems non-singular. How can I (algebraically) check that the whole intersection is non-singular? $\endgroup$ – Abramo Jan 17 '14 at 11:09
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    $\begingroup$ Dear Abramo, the intersection of $Q$ and $C$ will be singular at a point if the differentials $dQ$ and $dC$ are linearly dependent at that point. So you need to calculate the differentials, find their dependency locus, and check that it does not intersect your curve. $\endgroup$ – user64687 Jan 17 '14 at 11:15
  • $\begingroup$ Ok, this is super clear, thanks! $\endgroup$ – Abramo Jan 17 '14 at 12:52

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