Computation for the cohomology of $\mathcal{O}_Y(k)$ where $Y$ is a conic in $\mathbb{P}^3_{\mathbb{k}}$

Question

Let $Y$ be a conic in $\mathbb{P}^3_{\mathbb{k}}$ with $\mathbb{k}$ algebraically closed.

I already know that $deg(Y)=2$, the genus of $Y$ is zero and $\omega_Y \cong \mathcal{O}_Y(-1)$.If $L$ is the divisor corresponding to $\mathcal{O}_Y(1)$ we know $deg(L)=2$ thus Riemann-Roch theorem states that $h^0(\mathcal{O}_Y(k))-h^1(\mathcal{O}_Y(k))=2k+1$.

Serre duality gives $h^i(\mathcal{O}_Y(k))$ for $i=1,2$ and $k=-1,0$ since $H^0(\mathcal{O}_Y)=\mathbb{k}$.

My question is : Is everything correct? And is it possible to compute $h^i(\mathcal{O}_Y(k))$ for every other $k$?

Answer 1

A slight expansion of my comment: any conic over an algebraically closed field is isomorphic to $\mathbb P^1$, and $\mathcal O_{\mathbb P^3}(k)$ restricts to $\mathcal O_{\mathbb P^1}(2k)$. The global sections here can all be computed by counting binary forms of the relevant degree, and if needed $h^1$ can be computed from $h^0$ by applying Serre duality: $h^1(\mathcal O_{\mathbb P^1}(n)) = h^0(\mathcal O_{\mathbb P^1}(-2-n))$.