I'm only looking for some references. So I understand that if $C$ is a smooth curve of genus $g>1$, we have an injection $$C^{(2)}\rightarrow J(C),$$ $$(P,Q)\mapsto [P+Q-D_0]$$ where $C^{(2)}$ is a symmetric product and $D_0$ is an arbitrary but fixed divisor in $Div(C)$ of order 2. I have also seen somewhere that this map is an isomorphism if we exclude the zero divisor.
I only want to find the proof of these 2 facts somewhere in literature since I can't seem to prove it myself, but all the proofs I was able to find were too general and I couldn't understand them.