 Principal polarization

A principal polarization of an abelian variety $A$ is a polarization of an abelian variety $\lambda :A\to A^{\vee }$ that is an isomorphism. Equivalently, its isogeny has degree one. It identifies $A$ with its dual abelian variety through positivity supplied by an ample line bundle; an arbitrary isomorphism with the dual need not be a polarization of an abelian variety.^[ref]

For an elliptic curve $E$ with origin $0$, the degree-one line bundle ${\mathcal{O}}_E(0)$ gives a principal polarization. Replacing it by ${\mathcal{O}}_E(n\cdot 0)$, for positive $n$, multiplies the corresponding map to the dual by $n$ and gives degree $n^2$. Thus positivity alone does not make a polarization of an abelian variety principal: the size of its kernel matters.