I am trying to do an exercise from Rick Miranda's Book that is the following :
Let $X$ be an algebraic curve $g=2$ and $K$ a canonical divisor, show that $\phi_{2K}$ maps $X$ to a smooth projective plane conic.
Now first thing we need to do is to see that dim$L(K)=3$ but this is doable thanks to Riemann-Roch and the fact that $deg(K)=2>0$. Now what we would to see I think is that in fact the image of this map is a Riemann-Surface and for this we need to see that for every $p\in X$ we have that $dim(K-2p)=dimL(K)-2$, and this I cannot prove, my attempt was to use Riemann-Roch and we would get that $dim L(K-2p)=dimL(-K+2p)+1$, and suppose that $dim L(-K+2p)=1$ then we would have that $2p=div(fw)$ but I cant seem to get a contradiction with this.
New edit : I was able to get a contradiction with this because since $g=2$ then $X$ is an hyperilliptic curve and we know how holomorphic $1$-forms are supposed to behave and we cannot have something like $2p$, and so the image will in fact be a Riemann surface.
Also assuming that we know the image is a Riemann surface we know that there are 3 possibilities since $deg(2K)=4$ and I was able to see why $deg(\phi_{2k})=1$ and $deg(Y)=4$ is impossible thanks to the Hurwitz Formula, but I cant seem to discard the case $deg(\phi_{2k})=4$ and $deg(Y)=1$.
Any help is aprecciated, Thanks in advance!