On the Invariant Theory of $ \mathbb{G}_{a} $-Actions from a Geometric Perspective
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918144162922496 |
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| author | Maguire, Stephen |
| author_facet | Maguire, Stephen |
| contents | In this paper we give a strict classification of $ \mathbb{G}_{a} $-representations. This is done through the notion of a $ c(t) $-pair. Namely if $ \operatorname{Spec}(A) $ is a $ \mathbb{G}_{a} $-variety with action $ β$, then a $ c(t) $-pair is a pair of elements $ (g,h) $ such that $ g(t_{0} \ast x) = g(x)+c(t_{0}) h(x) $. This allows us to describe exactly when an affine, $ \mathbb{G}_{a} $-stable, sub-variety $ D(h) $ is a trivial bundle over $ D(h)//\mathbb{G}_{a} $. If $ \operatorname{Spec}(A) $ is a $ \mathbb{G}_{a} $-variety, we define the large pedestal ideal $ \mathfrak{P}_{g}(A) $ and the pedestal ideal $ \mathfrak{P}(A) $. If $ β: \mathbb{G}_{a} \to \operatorname{GL}(\mathbf{V}) $ is a $ \mathbb{G}_{a} $-representation, then we classify such a representation on whether: a) the large pedestal ideal $ \mathfrak{P}_{g}(S_{k}(\mathbf{V}^{\ast})) $ is equal to zero. b) the large pedestal ideal is non-zero, but the pedestal ideal is equal to zero. or c) the pedestal ideal is non-zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Invariant Theory of $ \mathbb{G}_{a} $-Actions from a Geometric Perspective Maguire, Stephen Algebraic Geometry 14D24 In this paper we give a strict classification of $ \mathbb{G}_{a} $-representations. This is done through the notion of a $ c(t) $-pair. Namely if $ \operatorname{Spec}(A) $ is a $ \mathbb{G}_{a} $-variety with action $ β$, then a $ c(t) $-pair is a pair of elements $ (g,h) $ such that $ g(t_{0} \ast x) = g(x)+c(t_{0}) h(x) $. This allows us to describe exactly when an affine, $ \mathbb{G}_{a} $-stable, sub-variety $ D(h) $ is a trivial bundle over $ D(h)//\mathbb{G}_{a} $. If $ \operatorname{Spec}(A) $ is a $ \mathbb{G}_{a} $-variety, we define the large pedestal ideal $ \mathfrak{P}_{g}(A) $ and the pedestal ideal $ \mathfrak{P}(A) $. If $ β: \mathbb{G}_{a} \to \operatorname{GL}(\mathbf{V}) $ is a $ \mathbb{G}_{a} $-representation, then we classify such a representation on whether: a) the large pedestal ideal $ \mathfrak{P}_{g}(S_{k}(\mathbf{V}^{\ast})) $ is equal to zero. b) the large pedestal ideal is non-zero, but the pedestal ideal is equal to zero. or c) the pedestal ideal is non-zero. |
| title | On the Invariant Theory of $ \mathbb{G}_{a} $-Actions from a Geometric Perspective |
| topic | Algebraic Geometry 14D24 |
| url | https://arxiv.org/abs/2509.15438 |