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This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one.

Given $X \subset \mathbb{CP}^N$ an abelian variety of dimension $n$ such that $N$ is the minimal dimension of embedding (see this question, generically $N = 2n+1$ for $n \geq 3$), what are the homogeneous polynomials that cut out $X$, and how many are required?

From what I've learned so far, unless $n=1$, $X$ is not a complete intersection. Cubic polynomials are always enough to cut out $X$, according to a comment deep into the MSE post in which this question originated. An abelian surface embeds in $\mathbb{CP}^4$ if and only if it is the zero-locus of some (generic) section of the Horrocks-Mumford bundle, if I understand correctly.

In particular, I'm interested in the case of dimension $2$, i.e. abelian surfaces, at the very least. Let $X$ be an abelian surface arising as the zero-locus of a section of the Horrocks-Mumford bundle over $\mathbb{CP}^4$, and let $Y$ be an abelian surface which embeds in $\mathbb{CP}^5$ but no less.

What are explicit, simple examples of families of homogeneous polynomials that cut out $X \subset \mathbb{CP}^4$ and $Y \subset \mathbb{CP}^5$?

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Horrocks-Mumford surfaces are cut out in ${\mathbb P}^4$ by 3 quintic and 15 sextic polynomials; the equations will have many dependencies (syzygies) between them. The references I found are [Manolache, Syzygies of abelian surfaces..., J für die reine und angewandte Mathematik 384, 180-191. Theorem 1] and [Aure et al, Syzygies of abelian and bielliptic surfaces..., https://arxiv.org/abs/alg-geom/9606013, Corollary 3.3]. Both of these papers contain plenty of representation theory, which organises matters somewhat. As explained in the introduction of the latter paper (and clear in many ways), finding an abelian surface in low-dimensional projective space is rather an "accident" and the equations are not going to be "simple". There are more "natural" families in higher-dimensional spaces, where the equations organise somewhat better; see [Gross and Popescu, Equations of (1,d)-polarized Abelian Surfaces, https://arxiv.org/abs/alg-geom/9606013] and references therein.

Computer algebra allows one to make explicit (if not simple) calculations. The particular case of Horrocks-Mumford surfaces is treated by macaulay2 in [Eisenbud et al, Computations in algebraic geometry with Macaulay 2, Example 7.2.].

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  • $\begingroup$ Thank you for the answer, and sorry for the very late response. I guess I had assumed that any embedding of an abelian variety could be cut out by cubics. But from what you say, one should only see cubics when the embedding is in a much larger dimensional projective space, correct? I.e. one cannot have both a minimal dimension for the embedding AND have only cubics to cut out the locus? $\endgroup$
    – Paul Cusson
    Commented Jan 16 at 7:41
  • $\begingroup$ In particular, I'm curious as how to interpret this comment by Damian in the linked MSE post: "The theory developped in the articles of D. Mumford, "On the equations defining abelian varieties I,II,II" in principle allows an explicit computation of equations cutting out abelian varieties in some projective spaces. In particular, he shows that quadratic and cubical equations suffice." $\endgroup$
    – Paul Cusson
    Commented Jan 16 at 7:42
  • $\begingroup$ It is a basic result of a first projective geometry course that any projective variety can be described by (usually many) quadratic equations in a suitable (usually large) projective space. But this may be a very inefficient embedding, though possibly natural, depending on the situation. Just think of a hypersurface of some degree in projective space: it can be described by a single equation of large degree, or many equations of smaller degrees (=2). There is often a trade-off between the codimension and the degrees of the defining equations... $\endgroup$
    – Balazs
    Commented Jan 16 at 10:54
  • $\begingroup$ ...in the specific case of abelian varieties, I believe Mumford studies complete linear systems on an abelian variety A: you take an ample divisor L, and describe an embedding of A into projective space by taking all sections of L. Explicit sections are theta functions, and these satisfy the quadric and cubic relations Damian is talking about. Here we will have large codimension, and small degrees... $\endgroup$
    – Balazs
    Commented Jan 16 at 10:58
  • $\begingroup$ ...and then, somewhat accidentally, some of these abelian varieties will admit embeddings into smaller dimensional spaces by taking a subset of all sections (geometrically, a projection); but then the equations get more complicated. This must be the situation with the Horrocks-Mumford sections. $\endgroup$
    – Balazs
    Commented Jan 16 at 10:59

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