3 added 555 characters in body

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)$$_{\nu \in \mathbb N} so that A becomes normal? Examples: a) If \mathfrak m_\nu= \mathbb C [x,y] for all \nu we have \hat P \cong \mathbb A^2 \times \mathbb G_a. b) If \mathfrak m_\nu =(x^m,y^n)^\nu, we have A = \mathbb C [x,y,x^mt, y^nt]. In the second example, one can see that A is normal for instance if \mathfrak m_\nu=(x^2,y)^\nu, but not if \mathfrak m_\nu=(x^2,y^2)^\nu -- in the latter case I think the normalization would be defined by the sequence \mathfrak m_\nu=(x^2,xy,y^2)^\nu. Since A = \mathbb C [x,y,x^mt, y^nt]\cong\mathbb C[x,y,u,v]/(x^mv-y^nu) in example b), Sp(A) is a hypersurface in \mathbb C^4, so normality is equivalent to singularities being of codimension at least two. Edit: If the question is difficult in general, I am also interested in the following special case: For which monomial ideals \mathfrak m\subset\mathbb C[x,y], is the ring defined by the sequence \mathfrak m^\nu normal? In this situation I would expect something like: A is normal iff the support of \mathfrak m^\nu consists of all lattice points in the convex hull of the support of \mathfrak m^\nu in \mathbb R^2. Here the support of \mathfrak m^\nu is the set of pairs (k,l)\in\mathbb N^2 such that x^ky^l\in\mathfrak m^\nu. 2 deleted 1 characters in body Let \mathbb G_a denote the additive group of complex numbers. Definition: Let V \subset Y be a dense open subset of the affine variety Y and \pi : P \longrightarrow V a \mathbb G_a-principal bundle. An affine extension is an affine \mathbb G_a-variety \hat P together with a morphism \hat \pi: \hat P \longrightarrow Y and an equivariant open embedding \iota: P \hookrightarrow \hat P, such that the diagram$$ \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], $$with a decreasing sequence (\mathfrak m_\nu), {\nu \in \mathbb N} N, of ideals \mathfrak m\nu m_\nu \subset \mathbb C [x,y], such that 1. \mathfrak m_\nu \cdot \mathfrak m_\lambda \subset \mathfrak m_{\nu + \lambda} for all \nu, \lambda\in \mathbb N, 2. \mathfrak m_0= \mathbb C [x,y], and 3. V(\mathfrak m_\nu) \subset \mathbf o for \nu > 0. On the other hand, every finitely generated \mathbb C-algebra of that form defines an affine extension of the trivial bundle. Question: Does somebody know of a criterion on the sequence (\mathfrak m_\nu)$$_{\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.

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# 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.

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.