Abstract
The configuration space \(\mathcal{C}^{n}(X)\) of an algebraic curve X is the algebraic variety consisting of all n-point subsets Q ⊂ X. We describe the automorphisms of \(\mathcal{C}^{n}(\mathbb{C})\), deduce that the (infinite dimensional) group \(\mathrm{Aut}\,\,\mathcal{C}^{n}(\mathbb{C})\) is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of \(\mathcal{C}^{n}(\mathbb{C})\) are also computed. We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level.
2010 Mathematics Subject Classification: 14R20, 32M17.
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Notes
- 1.
The upper index will usually mean the dimension of the variety.
- 2.
The rank of an affine algebraic group is the dimension of its maximal tori.
- 3.
The complex Weyl chamber of type B studied in [35] is isomorphic to \(\mathcal{C}^{n}(\mathbb{C}^{{\ast}})\).
- 4.
See [18] for further examples of non-cancellation.
- 5.
The authors thank S. Kaliman for a useful discussion, where the latter observation appeared.
- 6.
Since D n is homogeneous, any hypersurface D n (Q) = c ≠ 0 is isomorphic to \(\mathcal{S}\mathcal{C}^{n-1}\).
- 7.
See Sect. 4.5 below.
- 8.
Due to a result of Iitaka [16, Proposition 5] the latter assumption holds if the regular locus \(\mathrm{reg}\,\mathcal{X}\) is of non-negative logarithmic Kodaira dimension. Moreover, in this case \(\mathrm{Aut}_{0}\,\mathcal{X}\) is an algebraic torus.
- 9.
In the Arxive version of this paper we provide a proof of this theorem conformal to our notation.
- 10.
This is not a stratification of \(\mathrm{sing}\,\Sigma ^{n-1}\) since \(\Sigma _{\mathrm{Maxw}}^{n-2} \cap \Sigma _{\mathrm{cau}}^{n-2}\neq \varnothing \).
- 11.
In view of Theorem 8.2 , for n > 4 any non-Abelian holomorphic endomorphism of \(\mathcal{C}_{\mathrm{blc}}^{n-1}\) is surjective.
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Lin, V., Zaidenberg, M. (2014). Configuration Spaces of the Affine Line and their Automorphism Groups. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_24
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