I'm getting confused on an exercise in Appendix B of Hartshorne's Algebraic Geometry:
If two affine nonsingular curves are isomorphic as complex manifolds, then they're isomorphic as algebraic varieties.
I don't know how the affine and nonsingular condition can be used here. Any proof or reference is appreciated.
The field of rational functions are isomorphic as well as the ring of regular functions.
The non-singular requirement is to make sure the ring of regular functions is the same on both side.
An isomorphism of affine variety is the same as an isomorphism of their coordinate ring.
For example $x $ is in $\Bbb{C}[x]$ but not in $\Bbb{C}[x^2,x^3]$ ($\cong \Bbb{C}[y,z]/(y^3-z^2)$ the coordinate ring of $V(y^3-z^2)$) whereas in both case the function field is $\Bbb{C}(x)$ and the underlying complex manifold is $\Bbb{C}$.
Thus there is no affine variety isomorphism $\Bbb{A^1_C} \to V(y^3-z^2)$. This is because $V(y^3-z^2)$ is singular at $(0,0)$.
From the isomorphic field of rational functions you can construct a common compact Riemann surface $C$ (isomorphic to a non-singular projective curve) containing both $A,B$, that $A,B$ are non-singular implies that their coordinate ring is the whole ring of meromorphic functions on $C$ whose poles are located on $C-A,C-B$.