- #1

- 116

- 1

## Summary:

- connectedness of the projective algebraic curve

## Main Question or Discussion Point

I am following the proof to show that the complex torus is the same as the projective algebraic curve.

First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##.

Second we make an extension and add the point removed above from the torus. But now the maps goes into ##\mathbb{P}^2## which is in ##\mathbb{C}^3##.

The extension sends the missing point of the torus coming from point #z=0# in the lattice to the point ##[0,1,0]## in ##\mathbb{P}^2##, the so called point of infinity.

But I don't see and cannot imagine why the map constructed this way is continuous after we add a point of infinity to a continuous curve in ##\mathbb{P}^2##. Also related, when we "compactify" and add a point of infinity to our connected affine algebraic curve in ##\mathbb{C}^2##, is the resulting projective algebraic curve in ##\mathbb{C}^3## connected too?

The proof which I am following doesn't check the continuity of the extended map and connectedness of the projective algebraic curve explicitly.

Do we make the conclusion about the connectedness of the projective algebraic curve from establishing a biholomorphic map with the complex torus ?

Am I missing something either intuitive or a proved earlier result?

First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##.

Second we make an extension and add the point removed above from the torus. But now the maps goes into ##\mathbb{P}^2## which is in ##\mathbb{C}^3##.

The extension sends the missing point of the torus coming from point #z=0# in the lattice to the point ##[0,1,0]## in ##\mathbb{P}^2##, the so called point of infinity.

But I don't see and cannot imagine why the map constructed this way is continuous after we add a point of infinity to a continuous curve in ##\mathbb{P}^2##. Also related, when we "compactify" and add a point of infinity to our connected affine algebraic curve in ##\mathbb{C}^2##, is the resulting projective algebraic curve in ##\mathbb{C}^3## connected too?

The proof which I am following doesn't check the continuity of the extended map and connectedness of the projective algebraic curve explicitly.

Do we make the conclusion about the connectedness of the projective algebraic curve from establishing a biholomorphic map with the complex torus ?

Am I missing something either intuitive or a proved earlier result?