Zero Sum of a Theta Section

Choose a linear combination of the three degree $3$ theta basis sections: \(F(z)=a_0s_0(z)+a_1s_1(z)+a_2s_2(z).\)

For a generic choice of $a_0,a_1,a_2$, this section has three zeros: \(z_1,\quad z_2,\quad z_3.\)

The theta transformation rule fixes their sum on the torus. After normalizing the origin used by the embedding, this becomes \(z_1+z_2+z_3=0 \quad \text{in } \mathbb C/\Lambda.\)

Changing $a_0,a_1,a_2$ changes the line in projective space. The three zeros move, but their normalized sum stays zero. Without that normalization, their raw sum is a fixed point of $\mathbb C/\Lambda$.