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Why smooth projective curves are connected? I actually want to show that they are connected, compact, 2-manifolds. (projective algebraic curves are zeros of a homogenous polynomial of three variables with complex coefficients in CP2. We will call a projective algebraic curve smooth in point p if its gradient is not zero at point p.)

Ahmadreza Khazaeipoul
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  • What is your definition of a projective curve? My definition would not require them to be connected. – hunter Oct 07 '20 at 04:08
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    Can you show they are branched coverings of the projective line? – Angina Seng Oct 07 '20 at 04:51
  • It depends on your definitions. Some authors require all varieties to be irreducible so curves are automatically irreducible and therefore connected. Without that convention, e.g. $\mathbb{P}^1 \sqcup \mathbb{P}^1$ is a disconnected smooth projective curve. – Qiaochu Yuan Oct 07 '20 at 06:25
  • I think any smooth projective curves are irreducible due to Bezout's theorem. – Ahmadreza Khazaeipoul Oct 07 '20 at 08:58

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