I have a general question with a specific application. The cohomologies of line bundles on $\mathbb{P}^n$ are known and in particular, $H^0(\mathbb{P}^n, \mathcal{O}(d))$ is canonically isomorphic to the space of homogeneous polynomials in the variables $x_0, \ldots, x_n$ of degree d. Note that the $x_i$ are the projective coordianates. If I now have a projective variety $V := V(f(x_0, \ldots,x_n) \subset \mathbb{P}^n$, then is there any way to describe $H^0(V, \mathcal{O}(d))$?
In particular, I am trying to explicitly see what the space $H^0(\Sigma_2, \mathcal{O}(1,0))$ is where $\Sigma_2$ is the second Hirzebruch surface defined by $$\Sigma_2 = \left\{ ([y_0, y_1,y_2], [x_0,x_1])\vert x_0^2 y_1 = x_1^2 y_2\right\}\subset \mathbb{P}^2\times \mathbb{P}^1$$ And $$\mathcal{O}(p,q):= \pi_1^*(\mathcal{O}_{\mathbb{P}^2}(p)) \otimes \pi_2^*(\mathcal{O}_{\mathbb{P}^1}(q)).$$ I know that it is four-dimensional and three generators are given by the coordinate functions $y_0, y_1, y_2$. But what's the fourth? Thanks for the help with my general or specific question.