Let $F\in \mathbb C[z_0, z_1,z_2, z_3]$ be a homogeneous polynomial of degree $d$ and let $X := \{x \in \mathbb CP^3 \ | \ F([x]) = 0\}$.
I have read that if $d>4 $ then the canonical bundle $K_X$ is ample and I would like to undestand why.
By adjunction, $K_X = (d-4) H_X$ where $H_X$ is the restriction of the line bundle $H$ associated to an hyperplane in $\mathbb {CP}^3$. Consequently $K_X^2 = d (d-4)^2$. I also have computed $b_2 = d^3 −4d^2 +6d−2$ and $b^+ = \frac{(d-3)(d-2)(d-1)}{3}+1$.
I know of the Nakai-Mosheizon criterion: $K_X$ is ample iif $K_X^2 >0$ and $K_X \cdot C >0$ for all curves $C $ in X. I don't see however how I can verify the latter property. Maybe we can prove this starting from the definition of ample?
Thanks to Gunnar Þór Magnússon for suggesting the following approach.
Let $s_0,\dots, s_n \in Hol(X, L)$ be a basis of holomorphic functions. The map $i_L : X\to \mathbb P(Hol(X,L))^*\simeq \mathbb{CP}^{n}$ can be written using a local trivialization $U_\alpha$ of $L$ as $i_L(x) = [s_{0,\alpha}(x): \dots :s_{n,\alpha}(x)]\in \mathbb{CP}^n$.
Assume that L is very ample, then $i_L$ is an embedding.
If we consider $L|_Y\to Y$ where $Y\subset X$ is complex submanifold, then we can take as basis for $Hol(Y,L|_Y)$, $s_0|_Y,\dots, s_n|_Y, \dots, s_N$. Notice that it is possible that some of the sections coincide over $Y$. Nevertheless the map $i_{L|_Y}(y) = [s_{0,\alpha}(y): \dots :s_{N,\alpha}(y)]\in \mathbb{CP}^{N}$ will be an embedding because $(i_{L})|_Y$ is an embedding (even if some sections coincide over $Y$, there will be enough sections to separate the points of $Y$ thus the map will be injective, opennes follows in a similar manner).
Consequently $L$ (very) ample implies that $L|_Y$ is (very) ample too.
Now returing to the original problem, it is enough to prove that $H = \mathcal{O}(1)\to \mathbb {CP}^3$ is ample (which will imply that $kH$ is ample for $k>0$). We note that $H$ is very ample, indeed the holomorphic sections induced by $z_0,z_1, z_2, z_3\in \mathbb{C}[z_0,\dots, z_3]$ induce an the inclusion $i_H([z_0:\dots, z_3]) = [z_0:\dots: z_3]\in \mathbb{CP}^3$.