On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne):


1. We note that the map to its Jacobian given by $\mathcal{O}(p - p\_0)$ for a fixed point $p\_0$ is an isomorphism; ergo it inherits a group structure from the Jacobian.
2. In fact, if we embed it into $\mathbb{P}^2$ via the linear system $|3p\_0|$, then three colinear points have $p + q + r \sim 3p\_0$ and so this is in fact the group law inherited from Pic0.


Is there an analogous way to do this for Abelian varieties? In Lange & Birkenhake they simply define an Abelian variety to be $\mathbb{C}^n$ modulo a lattice, and so it automatically comes with a group structure. Still, this seems unsatisfying in comparison to the way we can do so for elliptic curves.


That being said, the previous method doesn't seem to work for Abelian varieties; divisors no longer correspond to formal sums of points and so any comparison with Pic0 wouldn't obviously yield a group structure on the points of X.


To make matters worse, the map that I tend to think of which takes X to Pic0(X) is given by $p \mapsto t\_p^\*L \otimes L^{-1}$ for a given line bundle $L$ on X, where $t\_p : X \to X$ is the map... defined by translation in X. So this map already requires the group structure on X to be defined.


So is there a way of defining the group law analogous to that of an elliptic curve?


NB: I do note that an elliptic curve can be defined as the zero locus of a cubic equation in $\mathbb{P}^2$, where I'm not sure how else we might define an Abelian variety other than as $\mathbb{C}^n$ modulo a lattice, and so perhaps the question is moot.



Is there an intrinsic way to define the group law on Abelian varieties?
Any torsor $V$ under an abelian variety over any field $K$ is caonically isomorphic to its degree $1$ Albanese variety $\operatorname{Alb}^1(V)$, which is itself a torsor under the degree $0$ Albanese variety $\operatorname{Alb}^0(V)$. Note that the $\operatorname{Alb}^0$ of any smooth projective variety is a complete, geometrically connected group variety. 


Assuming there exists a $K$-rational point $O$, one can subtract $O$ to obtain an isomorphism $\operatorname{Alb}^1(V) \stackrel{\sim}{\rightarrow} \operatorname{Alb}^0(V)$. Pulling back via the composite of these isomorphisms puts a group structure on $V$ depending only on the chosen base point $O$. 


