Automorphisms of the plane preserving a curve

Blanc, Jérémy; Stampfli, Immanuel

doi:10.14231/ag-2015-009

Cited by 10 publications

(4 citation statements)

References 16 publications

(16 reference statements)

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“…The proof in [Lam02] uses blow-ups and contractions of the line L ∞ = P 2 \ A 2 , in the spirit of the methods used in this article. For more proofs with a similar strategy see [BD11] and [BS15].…”

Section: Theorem 37 ([Jun42] [Vdk53]mentioning

confidence: 99%

Isomorphisms between complements of projective plane curves

Hemmig¹

2019

Épijournal De Géométrie Algébrique

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In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.

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Section: Theorem 37 ([Jun42] [Vdk53]mentioning

confidence: 99%

Isomorphisms between complements of projective plane curves

Hemmig¹

2019

Épijournal De Géométrie Algébrique

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“…For a point E(q) ∈ E the local coordinate chart centered at E(q) is given by (14) x E(q) , y E(q) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (x p , y E − q), q ≠ ∞, 1 y E , y p , q = ∞ .…”

Section: Arcs and Multiplicitiesmentioning

confidence: 99%

“…where the coordinates (x 0 , y 0 ) near the fiber T * 0 ⊂ X 0 are chosen so that x 0 = µ * 0 (t), and y 0 = 0 defines the zero section of µ 0 ∶ X 0 → B. Regarding y 0 as a P 1 -coordinate on T 0 ⊂ X0 , we define local coordinates at each point T 0 (q) of T * 0 = T 0 ∩ X 0 via (14). For every i = 1, .…”

Section: Arcs and Multiplicitiesmentioning

confidence: 99%

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On automorphism groups of affine surfaces

Kovalenko¹,

Perepechko²,

Zaidenberg³

Advanced Studies in Pure Mathematics

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This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

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On the Automorphism Group of a polynomial differential ring in two variables

Baltazar¹,

Pan²

2021

Journal of Algebra

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Automorphisms of the plane preserving a curve

Cited by 10 publications

References 16 publications

Isomorphisms between complements of projective plane curves

Isomorphisms between complements of projective plane curves

On automorphism groups of affine surfaces

On the Automorphism Group of a polynomial differential ring in two variables

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