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This article describes the geometry of isomorphisms between complements ofgeometrically irreducible closed curves in the affine plane $\mathbb{A}^2$,over an arbitrary field, which do not extend to an automorphism of$\mathbb{A}^2$.  We show that such isomorphisms are quite exceptional. In particular, theyoccur only when both curves are isomorphic to open subsets of the affine line$\mathbb{A}^1$, with the same number of complement points, over any fieldextension of the ground field. Moreover, the isomorphism is uniquely determinedby one of the curves, up to left composition with an automorphism of$\mathbb{A}^2$, except in the case where the curve is isomorphic to the affineline $\mathbb{A}^1$ or to the punctured line $\mathbb{A}^1 \setminus \{0\}$. Ifone curve is isomorphic to $\mathbb{A}^1$, then both curves are in factequivalent to lines. In addition, for any positive integer $n$, we construct asequence of $n$ pairwise non-equivalent closed embeddings of $\mathbb{A}^1\setminus \{0\}$ with isomorphic complements. In characteristic~$0$ we evenconstruct infinite sequences with this property.  Finally, we give a geometric construction that produces a large family ofexamples of non-isomorphic geometrically irreducible closed curves in$\mathbb{A}^2$ that have isomorphic complements, answering negatively theComplement Problem posed by Hanspeter Kraft.. This also gives a negative answerto the holomorphic version of this problem in any dimension $n \geq 2$. Thequestion had been raised by Pierre-Marie Poloni.

Exceptional isomorphisms between complements of affine plane curves