I am having trouble understanding two statements:
Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ then $\hat{A}$ is a topologically a torus .
I understand up to here, as it follows from Riemann-Roch theorem. What I don't understand are the following statements:
It's (i.e. torus's) group of holomorphic automorphisms $Aut(\hat{A})$ is a complex Lie group whose connected component at the identity is a complex torus $T$ which acts freely and transitively on $\hat{A}$.
The quotient $\Gamma:= Aut(\hat{A})/T$ is a finite group of order at most six.
I don't understand either of these points. Could someone clarify? Which theorems are being used here? Please let me know of references, if possible, for the proof.
$\hat{A}$ is isomorphic (complex analytically) to a torus of the form $\mathbb{C}/\Lambda$, where $\Lambda=\langle 1,\tau\rangle$ is a discrete subgroup of rank 2 in $\mathbb{C}$ (that is, a lattice) and $\tau\in\mathbb{H}$. It is a well-known theorem that any holomorphic map between tori is the composition of a group homomorphism with a translation.
We see that translations are automorphisms of $\hat{A}$, and so $\hat{A}\leq\mbox{Aut}(\hat{A})$ (since if $t_x$ is translation by $x$, $t_x\circ t_y=t_{x+y}$). What's left is to look at the elements $G$ of the automorphism group that fix 0. An automorphism that fixes 0 is given by a complex number $\lambda$ such that $\lambda\Lambda=\Lambda$. It is not hard to show that this implies that $\lambda$ is either a fourth root or a sixth root of unity. After analyzing different cases, we get that $G$ can be either $\mathbb{Z}/2\mathbb{Z}$ (given by the involution $x\mapsto -x$, all tori have this), $\mathbb{Z}/4\mathbb{Z}$ (in this case the torus is isomorphic to the torus $\mathbb{C}/\langle1, i\rangle$) or $\mathbb{Z}/6\mathbb{Z}$ (and here the torus is isomorphic to $\mathbb{C}/\langle1,e^{2\pi i/6}\rangle$).
Putting all this together, we get that the automorphism group is isomorphic to $G\ltimes\hat{A}$, where $G$ has order 2 (this is true for the general torus), 4 or 6.
Edit: A good beginning reference for this is Miranda's book on Algebraic curves and Riemann surfaces.