Let $X$ be the quintic threefold in $\mathbb{CP}^{4}$ defined by the vanishing of the Fermat polynomial $$ x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}+x_{5}^{5} $$ and consider a rational curve $D$ on $X$. I would like to compute the Chern character $ch(\mathcal{O}_{D})$ of the structure sheaf of $D$. I believe that the answer should be $$ ch(\mathcal{O}_{D}) = -\frac{1}{5}[H]^{2} - \frac{1}{5}[H]^{3} $$ where $[H]$ is the (restriction of), the hyperplane class.
My first idea was to consider the short exact sequence $$ 0 \to\mathcal{I}_{D} \to \mathcal{O}_{X} \to \mathcal{O}_{D} \to 0 $$ where $\mathcal{I_{D}}$ is the ideal sheaf of $D$. Then by the additivity of the Chern character, we should have $$ ch(\mathcal{O}_{D}) = ch(\mathcal{O}_{X}) - ch(\mathcal{I}_{D}) $$ However, I have no idea how to identify the ideal sheaf $\mathcal{I}_{D}$. Any pointers would be greatly appreciated.
If $C$ is a curve of genus $g$ then $$ ch(\mathcal{O}_C) = [C] + (1 - g)[P] = \frac{d}5[H]^2 + \frac{1 - g}5[H]^3, $$ where $[P]$ is the class of a point. Indeed, the top term is $[C]$ by Grothendieck--Riemann--Roch, and the bottom term is determined by Hirzebruch--Riemann--Roch (note that the canonical class of $X$ is zero).