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Autour d'une surface rationnelle dans $\mathbb{C}^3$
Algebraic Geometry · Submitted 7 Dec 2005
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$.
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The abstract, sentence by sentence
Affine surfaces in $\mathbb{C}^{3}$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ have been increasingly studied during the past 15 years. Of particular interest is the fact that they come equipped with an action of the additive group $\mathbb{C}_{+}$ induced by such an action on the ambient space. The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in $\mathbb{C}^{3}$ with this property. In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of $\mathbb{C}^{3}$ which seems to be new.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Affine surfaces in $\mathbb{C}^{3}$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ have been increasingly studied during the past 15 years. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | Of particular interest is the fact that they come equipped with an action of the additive group $\mathbb{C}_{+}$ induced by such an action on the ambient space. | background 1.00 | 0.07 | 0.03 | asserted 1.00 |
| s2 | The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in $\mathbb{C}^{3}$ with this property. | gap 0.53 | 0.03 | 0.04 | hedged possibility 0.98 |
| s3 | In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$. | contribution 0.96 | 0.97 | 0.90 | asserted 1.00 |
| s4 | We give an elementary and self-contained proof of this fact. | method 0.84 | 0.93 | 0.05 | asserted 1.00 |
| s5 | As an application, we construct a wild but stably-tame automorphisme of $\mathbb{C}^{3}$ which seems to be new. | contribution 0.51 | 0.95 | 0.16 | hedged possibility 0.71 |
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The paper
Original abstract
Affine surfaces in $\mathbb{C}^{3}$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ have been increasingly studied during the past 15 years. Of particular interest is the fact that they come equipped with an action of the additive group $\mathbb{C}_{+}$ induced by such an action on the ambient space. The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in $\mathbb{C}^{3}$ with this property. In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of $\mathbb{C}^{3}$ which seems to be new.
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