Let $Y\subset\mathbb{P}^n$ be a smooth variety, and let $\epsilon:X = Bl_Y\mathbb{P}^n\rightarrow\mathbb{P}^n$ be the blow-up of $\mathbb{P}^n$ along $Y$.

Let $\widetilde{H}$ be the pull-back of the hyperplane section $H$ of $\mathbb{P}^n$, and $E$ be the exceptional divisor. If $H_Y =H\cdot Y$ we have
$$\widetilde{H}^{h-i}E^i = p^*H_Y^{n-i}\cdot i^*E^{i-1} = H_Y^{n-i}\cdot p_*i^*E^{i-1}.$$
Recall that $E = \mathbb{P}(N_{Y/\mathbb{P}^n})$, and $i^*E = -e$, where $e = c_1(\mathcal{O}_E(1))$. Let use denote by $s_j$ the Segre classes of $N_{Y/\mathbb{P}^n}$, and let $c = codim_{\mathbb{P}^n}(Y)$. We have the following intersection numbers:

- $\widetilde{H}^n = 1$;
- $\widetilde{H}^{n-i}\cdot E^i = 0$ for $i < c$;
- $\widetilde{H}^{n-i}\cdot E^i = (-1)^{i-1}s_{i-c}H_Y^{n-i}$ for $i\geq c$.

Let $C\subset\mathbb{P}^3$ be a smooth curve of degree $d$ and genus $g$. By the exact sequence
$$0\mapsto T_{C}\mapsto T_{\mathbb{P}^3|C}\rightarrow N_{C/\mathbb{P}^3}\mapsto 0$$
we get $s_1(N_{C/\mathbb{P}^3}) = -c_{1}(N_{C/\mathbb{P}^3}) = -4d-2g+2$. Then $\widetilde{H}^3 = 1$, $\widetilde{H}^2\cdot E = 0$, $\widetilde{H}\cdot E^2 = -s_0H_Y = -d$, and $E^3 = s_1 = 2-2g-4d$. For instance we can compute the cube of the anti-canonical divisor:
$$(-K_{Bl_C\mathbb{P}^3})^3 = (4\widetilde{H}-E)^3 = 64-12d+4d+2g-2 = 62-8d+2g.$$