Very Ample Line Bundle Example

Fix a complex torus \(E=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)\) and a very ample line bundle of degree $3$.

This line bundle has three independent holomorphic sections: \(s_0,\quad s_1,\quad s_2.\)

Explicitly, we can take the degree $3$ theta basis \(s_k(z)=\sum_{n\in\mathbb{Z}} \exp\left(3\pi i\left(n+\frac{k}{3}\right)^2\tau +6\pi i\left(n+\frac{k}{3}\right)z\right), \qquad k=0,1,2.\)

Choosing a point $z\in E$ determines three section values \(s_0(z),\quad s_1(z),\quad s_2(z).\)

These values become projective coordinates: \(z \mapsto [s_0(z):s_1(z):s_2(z)]\in\mathbb{P}^2.\)

In the picture, each section is represented by a theta function, and its phase is shown by color.