I need help to solve the following problem that appears on page 55 of the book of William Fulton entitled Algebraic Curves.
Exercise :
Let $ R = k[X,Y,Z] $, $F \in R$ an irreducible form of degree $n$, $ V=V( F ) \subset \mathbb{P}^2 $, and $ \Gamma = \Gamma_h (V) = k[X,Y,Z] / I(V) $ the homogeneous coordinate ring of $V$.
1) Construct an exact sequence $ \ 0 \to R\stackrel{\psi} \to R \stackrel{\varphi} \to \Gamma \to 0 $, where $ \psi $ is the multiplication by $F$.
2) Show that $ \mathrm{dim}_k \ \{ \text{forms of degree} \ d \ \ \text{in} \ \Gamma \} = dn - \dfrac{n(n-3)}{2} $ if $ d > n $.
Thanks in advance for your help.