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This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping my follow-up questions and related discussion are of interest. To give an idea of my weak background, I am currently still going through Griffiths and Harris.

To rehash, let $X = \mathbb{C}^n / \Lambda$ be a complex torus of complex dimension $n$, with the lattice $\Lambda$ satisfying the Riemann conditions. Knowing that this condition guarantees that $X$ embeds in $\mathbb{CP}^N$ for some $N$, there exists a family of homogeneous polynomials whose zero locus is the embedded torus.

My original question was to ask for explicit examples of such embedded tori (for $n > 1$, as the case of elliptic curves is well known) and their corresponding polynomials. After reviewing the recommended literature I still could not figure out even a simple example, though I'm sure some must be known and there should be a method to directly compute these polynomials given $\Lambda$. I did learn in Birkenhake-Lange's textbook that nothing more than cubics are enough.

But it was pointed out to me that there are some subtleties that would make this a difficult task, which lead to my updated questions:

Given $X$, what is the smallest integer $N$ such that $X$ embeds in $\mathbb{CP}^N$? And is this number $N$ independent of the choice of lattice $\Lambda$? I.e. does it only depend on the dimension $n$ of the torus?

Mention of complete intersection (which I don't know about yet) was brought up, and if I remember correctly, it can be used to argue that any complex dimension $2$ torus cannot embed in $\mathbb{CP}^4$.

At the very least, I would be happy with a reference for a polynomial description of the simplest non-trivial case, $n=2$, for some "easy" $\Lambda$, just to get me started.

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  • $\begingroup$ In the case of Jacobians of curves, you might want to look at the article "An explicit algebraic representation of the Abel map" by Greg W. Anderson at IMRN academic.oup.com/imrn/article/1997/11/495/665888 There is also a follow-up article in Adv. Math. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Oct 4, 2023 at 14:34
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    $\begingroup$ If $X$ is of dimension $g$ and polarised with polarisation of degree $d^2$, then $N$ can be taken to be $d\cdot 3^g-1$. This follows from the fact that the cube of an ample line bundle is very ample on an abelian variety, and from the Riemann-Roch theorem. $\endgroup$
    – Damian Rössler
    Commented Oct 4, 2023 at 16:23
  • $\begingroup$ @DamianRössler interesting, and is this $N$ known to be minimal if one can choose small $d$? $\endgroup$
    – Paul Cusson
    Commented Oct 4, 2023 at 18:14
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    $\begingroup$ " it can be used to argue that any complex dimension 2 torus cannot embed in CP4" That is not correct. There are abelian surfaces embedded in P^4, namely the zero-loci of sections of the Horrocks-Mumford bundle. It is true that a complete intersection in P^N cannot be a complex torus. $\endgroup$
    – Pop
    Commented Oct 4, 2023 at 21:37
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    $\begingroup$ mathoverflow.net/a/14183/65919 seems relevant $\endgroup$
    – Eoin
    Commented Oct 5, 2023 at 0:39

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