Exceptional isomorphisms between complements of affine plane curves

Blanc, Jérémy; Furter, Jean-Philippe; Hemmig, Mattias

doi:10.1215/00127094-2019-0012

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…It turns out that C \ L ⊂ P 2 \ L is then rectifiable. This is a consequence of the following proposition, proven in [BFH16,Proposition 3.16]. It also follows from the work of [KM83] and [Gan85] (see [BFH16, Remark 2.30]).…”

Section: A Very Tangent Linesmentioning

confidence: 55%

“…when the number k of complement points is 1) is of particular interest since the rigidity of Proposition 2.8 does not hold there. Indeed, by a result of P. Costa ([Cos12], [BFH16,Proposition A.3. ]), there exists a family of irreducible rational unicuspidal curves (C λ ) λ∈k * in P 2 that are pairwise projectively non-equivalent, but all have isomorphic complements.…”

Section: Conjecture 11 ([Yos84]mentioning

confidence: 99%

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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: A Very Tangent Linesmentioning

confidence: 55%

Section: Conjecture 11 ([Yos84]mentioning

confidence: 99%

Isomorphisms between complements of projective plane curves

Hemmig¹

2019

Épijournal De Géométrie Algébrique

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“…The following result generalises [BFH19, Theorem 1(2)], which is the case where the 1-dimensional schemes C, D are integral. We use for this the tools developed in [BFH19] (especially [BFH19, Proposition 3.16]). In the sequel we denote for any closed subscheme Z ⊂ A n the associated reduced scheme by Z red .…”

Section: 2mentioning

confidence: 99%

Automorphisms of the affine 3-space of degree 3

Blanc¹,

Santen²

2019

Preprint

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In this article we give two explicit families of automorphisms of degree ≤ 3 of the affine 3-space A 3 such that each automorphism of degree ≤ 3 of A 3 is a member of one of these families up to composition of affine automorphisms at the source and target; this shows in particular that all of them are tame. As an application, we give the list of all dynamical degrees of automorphisms of degree ≤ 3 of A 3 ; this is a set of 3 integers and 9 quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for n ≥ 1 all morphisms A 3 → A n of degree ≤ 3 with the property that the preimage of every affine hyperplane in A n is isomorphic to A 2 .

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“…At the time that the first version of the present paper was being written, the case of irreducible curves on C 2 was still wide open. But since then it has been solved, again in the negative, by Blanc, Furter and Hemmig in a remarkable paper [BFH16] in which they make use of totally different methods to find counterexamples to the Complement Problem in the case where n = 2.…”

Section: Introductionmentioning

confidence: 99%