# Which curves can be found on Abelian varieties?

We know that each genus 2 curve is embedded into its degree 1 Jacobian.

Under which conditions on $$C$$, $$A$$, $$g$$ and $$n$$ is it possible for a genus $$g$$ smooth curve $$C$$ to be embedded in an Abelian variety $$A$$ of dimension $$n$$?

What can be said in the case $$n=2$$?

And what if $$A$$ is a Jacobian (and e.g. $$n=2$$)?

• If an abelian variety is simple, it cannot contain a curve of genus smaller than its dimension: this follows easily by considering the restriction of global differential forms. If an abelian variety of dimension g contains a curve of genus g, then the abelian variety is isogenous to the jacobian of the curve: this follows easily by using the fact that the Jacobian of the curve is also the Albanese variety of the curve. – damiano Apr 15 '10 at 10:01
• (even though this is not clear from what i wrote above, you need the abelian variety to be simple also in the second part.) – damiano Apr 15 '10 at 10:09
• @damiano: even if it doesn't completely answer to the question (I don't know if it's actually possible to have a complete answer), it's a valuable comment: you could have posted it as an answer. – Qfwfq Apr 15 '10 at 10:57
• If a curve C maps to an abelian variety A, then the Jacobian J(C) also maps to A. This is another way of realizing damiano's remark. It also suggests how one might try to find curves on A in general. – mdeland Apr 15 '10 at 13:29
• You might be interested in Ernst Kani's paper "Elliptic curves on abelian surfaces". – Dan Petersen Apr 15 '10 at 21:27

For any $g >0$ there exists an abelian surface $A$ containing a smooth curve $C$ of genus $g$; the surface can be assumed to be simple if $g > 1$:

For $g=1$, one can just consider $C \times C$. For $g>1$, consider the generic abelian surface with a polarisation of type $(1, g-1)$. By the adjunction formula, a smooth curve in the linear system corresponding to the polarisation will have genus $g$. (Such abelian surfaces can be constructed by considering suitable quotients of principally polarised ones ($n=1$))

One gets abelian varieties of any dimension $>1$ by taking products. However, this does not produce a fixed abelian surface containing curves of all genera $g>1$ but it seems likely that such surfaces should exist. It would be interesting to know whether this is possible for the generic principally polarised surface, or more generally ppav of any dimension $n$.

• Regarding your last paragraph: if A is an abelian surface with End(A) = Z, then it contains smooth curves of very few genera. NS(A) is generated by the polarization L in this case, and the genus of a smooth curve in |nL| is n^2(L.L)/2 - 1. The abelian surfaces you ask about do exist however: certain (but not all) products of elliptic curves will do the trick. – Ari Shnidman Jun 26 '16 at 2:30

Gian Pietro Pirola in [Curves on generic Kummer varieties, Duke Math. J. Volume 59, Number 3 (1989), 701-708] proves a rigidity theorem for curves of genus $g\le q-2$ in the Kummer variety of a $q$-dimensional abelian variety. As a consequence, he shows that a generic abelian variety of dimension $q\ge 3$ does not contain hyperelliptic curves of any genus.

In his paper "Abelian surfaces with (1,2)-polarization" Barth proves that a smooth genus $3$ curve can be embedded into an Abelian surface if and only if it admits an elliptic involution, i.e. a map of degree $2$ onto an elliptic curve.

If $C$ has a map onto a smooth curve $B$, there is a surjective map $J(C)\to J(B)$ hence $C$ maps to $J(B)$ and it is probably often an embedding.

• The map from $C$ to $J(B)$ factors through the map from $C$ to $B$ and hence is almost never an embedding. – Torsten Ekedahl Apr 15 '10 at 15:38