# Normality condition on graded algebra

Let $\mathbb G_a$ denote the additive group of complex numbers.

Definition: Let $V \subset Y$ be a dense open subset of the affine variety $Y$ and $\pi : P \longrightarrow V$ a $\mathbb G_a$-principal bundle. An affine extension is an affine $\mathbb G_a$-variety $\hat P$ together with a morphism $\hat \pi: \hat P \longrightarrow Y$ and an equivariant open embedding $\iota: P \hookrightarrow \hat P$, such that the diagram

$$\begin{array}{ccc} P & \hookrightarrow & \hat P \\\ \downarrow & & \downarrow \\\ V & \hookrightarrow & Y \end{array}$$ is commutative and $\iota (P)= \hat \pi^{-1}(V).$

I am interested in affine extensions of the trivial $\mathbb G_a$-bundle over the affine plane punctured at the origin, i.e. $\mathbb A^2_*:=Sp(\mathbb C[x,y])\setminus\{\mathbf o\}$ with $\mathbf o:=(x,y)$, and I have the following description:

Proposition: If $P \longrightarrow \mathbb A_*^2$ is the trivial bundle, then any affine extension $\hat P \longrightarrow \mathbb A^2$ is of the form $$\hat P = Sp (A),$$ where $$A= \bigoplus_{\nu=0}^\infty \mathfrak m_\nu t^\nu \subset \mathbb C [x,y,t],$$ with a decreasing sequence $(\mathfrak m_\nu)$, $\nu \in \mathbb N$, of ideals $\mathfrak m_\nu \subset \mathbb C [x,y]$, such that

1. $\mathfrak m_\nu \cdot \mathfrak m_\lambda \subset \mathfrak m_{\nu + \lambda}$ for all $\nu, \lambda\in \mathbb N$,

2. $\mathfrak m_0= \mathbb C [x,y]$, and

3. $V(\mathfrak m_\nu) \subset \mathbf o$ for $\nu > 0$.

On the other hand, every finitely generated $\mathbb C$-algebra of that form defines an affine extension of the trivial bundle.

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