Let $C:F(x,y,z)=0$ be a curve given by irreducible homogeneous polynomial $F\in K[x,y,z]$, $\deg F>0$ then $C$ has infinitely many points over $\overline K$
Why is this true, If one finds one point (an appropriate) on the curve, then by dehomogenezing and using the fact that algebraic closure is always infinite one obtains infinitely many points, is this the reason ? Where is the irreducibility required ?
The zero set of $F$ is called curve, so $C$ cannot be affine, since an algebraic curve in 3-dimensional space cannot be described by just one equation. $C$ must be a (plane) projective curve. To prove that it has infinitely many points one can proceed in the way suggested in the post. The irreducibility is not needed.