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I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, if such a thing existed, would be nice. The more explicitly such a curve can be described, the better.

The only way I know how to produce curves in $A$ is the general method of embedding it into a projective space and taking sections with hyperplanes. I expect this will probably give curves of quite large genus. Are there more specialized ways to produce curves in the setting of abelian surfaces? Are there nice examples of known simple curves?

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    $\begingroup$ Related: mathoverflow.net/questions/21439/… $\endgroup$ – Qfwfq Jan 22 '19 at 0:39
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    $\begingroup$ The Jacobian of a genus $2$ curve is an abelian surface, and you have the Abel-Jacobi map $\endgroup$ – Qfwfq Jan 22 '19 at 0:41
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The answers in the linked question above more or less give the current state of affairs at this level of generality. Let me just add that as you say, by Riemann-Roch you can obtain an upper bound for the genus of a complete intersection curve in a principally polarised $(A,L)$ (as e.g. $L^g=g!$ and $3L$ is very ample). If $A$ is not principally polarised, from Zarhin's trick $(A\times A^\vee)^4$ will be so, and one can push down curves under the first projection to obtain a similar upper bound. I have not seen that one can produce a better upper bound than this in this generality.

As mentioned, Pirola has proved that a generic complex abelian variety of dimension at least 3 does not contain a hyperelliptic curve, and this has been generalised to arbitrary characteristic by de Jong and Oort.

As far as more general lower bounds are concerned, very recently, Voisin has proven a logarithmic lower bound on the gonality of a curve contained in a very general abelian variety (equivalently a curve covering the abelian variety), and conjectures that if $\dim A\geq 2k-1$ then the gonality of any curve contained in $A$ is $\geq k+1$.

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