Let $C$ be an irreducible conic in $\mathbb{P}^2$ (in order to fix ideas we can take $C = \mathbb{V}( x_0^2 + x_1^2 + x_2^2)$ where $x_0, x_1, x_2$ are the homogeneous coordinates in $\mathbb{P}^2$. Let $X = \mathbb{P}^2 \setminus C$. We know (by using degree-2 Veronese embedding) that $X$ in an affine variety.
My porpuse is to build a regular non-constant function on X, or, more generally, compute the ring of regular function $O_{X}(X) = \mathbb{C}[X]$.
I know that such a regolar function actually exists and I know it is essentially a polinomial function. But how to be sure to do the right choice? For example, could be the function $f= x_0$ work?
I'm not sure, but $X$ should be isomorphic to $V_{2,2} \cap \mathbb{A}^5$. Is this true? ($V_{2,2} = v_2 ( \mathbb{P}^2)$)
If it is, then i can compute $V_{2,2} \cap \mathbb{A}^5$ as the zero locus of an ideal by substituting $a_0 = 1$ in the equations of $V_{2,2}$ if we denote $a_i$ the homogeneous coordinates in $\mathbb{P}^5$. Let me call $I$ this ideal. Then i can compute "easly" $\mathbb{C}[a_0, \dots, a_5] / I$. Right?
Recap:
- Is it true that $X$ is isomorphic to $V_{2,2} \cap \mathbb{A}^5$?
- $\mathbb{C}[a_0, \dots, a_5] / I$ is isomorphic to something easier to understand?