Let $F$ be a non constant holomorphic map between two compact Riemann surface $X,Y$. Assume for each point $p\in X , mult_p(F)=1$. Then ,what can we say about the degree of that map (deg(F))?
Locally $F$ looks like a coordinate chart function. Can we say $F$ is an isomorphism? I am reading "Riemann Hurwitz formula " this question came in my thought but unable to get answer.
First off, the multiplicity is usually only defined for non-constant holomorphic mappings. So, let me assume that $F\colon X\rightarrow Y$ is a non-constant holomorphic mapping between compact Riemann surfaces.
We have that $F$ is surjective, since $F(X)$ is a non-empty clopen subset of the connected topological space $Y$. Thus, if we knew that $F$ were injective, we would be done. Now, since $F$ is unramified, by looking at its Local Normal Form, we see that $F$ is injective locally. However, in general, $F$ is not injective globally.
Let me give an example. By the Riemann-Hurwitz formula, any non-constant holomorphic mapping between complex tori is unramified. There are, however, holomorphic mappings between complex tori which are not biholomorphisms (i.e. that do not have degree one).
To see this, take you favourite (discrete) lattice $\Gamma \subset \mathbb{C}$, say the Gaussian integers $\Gamma=\mathbb{Z}+\mathbb{Z}i$. Pick your favourite complex number $\alpha \in \mathbb{C}\setminus\{0\}$, say $\alpha=1+i$. Then take a lattice $\Gamma'$ with $\alpha\Gamma\subsetneq \Gamma'$, say $\Gamma'=\mathbb{Z}\frac{1+i}{2}+\mathbb{Z}\frac{-1+i}{2}$. Next, consider the mapping $$f_\alpha\colon\mathbb{C}/\Gamma\longrightarrow\mathbb{C}/\Gamma'; [z]_\Gamma\longmapsto [\alpha\cdot z]_{\Gamma'}.$$ This mapping is well-defined since $\alpha\Gamma\subseteq\Gamma'$. Moreover, it is a non-constant group homomorphism. Additionally, it is holomorphic (why?). Finally, it is non-injective since $\alpha\Gamma\subsetneq\Gamma'$. Namely, pick a complex number $z\in\Gamma'\setminus\alpha\Gamma$. Then $f_\alpha([\frac{1}{\alpha}z]_\Gamma)=[0]_{\Gamma'}$, but $\frac{1}{\alpha}z\notin \Gamma.$ Thus, the kernel of $f_\alpha$ is non-trivial.
The example I gave is of minimal genus. Namely, by the Riemann-Hurwitz formula any unramified non-constant holomorphic mapping $f\colon X\rightarrow Y$ between compact Riemann surfaces satisfies $2g(X)-2=(2g(Y)-2)\operatorname{deg}(f)$, where $g(-)$ denotes the genus. If $\operatorname{deg}(f)\geq 2$, then $g(X)\geq 1$ and $g(Y)\geq 1$.
Note that Aphelli's example in the comments is a special case of the above construction.