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.

**Question:** Does somebody know of a criterion on the sequence $(\mathfrak m_\nu)$$_{\nu \in \mathbb N}$ so that $A$ becomes normal?

**Examples:**
a) If $\mathfrak m_\nu= \mathbb C [x,y]$ for all $\nu$ we have $\hat P \cong \mathbb A^2 \times \mathbb G_a$.

b) If $\mathfrak m_\nu =(x^m,y^n)^\nu$, we have $ A = \mathbb C [x,y,x^mt, y^nt]. $

In the second example, one can see that $A$ is normal for instance if $\mathfrak m_\nu=(x^2,y)^\nu$, but not if $\mathfrak m_\nu=(x^2,y^2)^\nu$ -- in the latter case I think the normalization would be defined by the sequence $\mathfrak m_\nu=(x^2,xy,y^2)^\nu$. Since $A = \mathbb C [x,y,x^mt, y^nt]\cong\mathbb C[x,y,u,v]/(x^mv-y^nu)$ in example b), $Sp(A)$ is a hypersurface in $\mathbb C^4$, so normality is equivalent to singularities being of codimension at least two.

**Edit:** If the question is difficult in general, I am also interested in the following special case: For which monomial ideals $\mathfrak m\subset\mathbb C[x,y]$, is the ring defined by the sequence $\mathfrak m^\nu$ normal? In this situation I would expect something like: $A$ is normal iff the support of $\mathfrak m^\nu$ consists of all lattice points in the convex hull of the support of $\mathfrak m^\nu$ in $\mathbb R^2$. Here the support of $\mathfrak m^\nu$ is the set of pairs $(k,l)\in\mathbb N^2$ such that $x^ky^l\in\mathfrak m^\nu$.