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Examples of plane curves admitting the same Galois closure for two projections. (English) Zbl 1495.14040

Let \(C\subset \mathbb{P}^{2}\) be an irreducible plane curve of degree \(d\geqslant 4\) over an algebraically closed field \(k\) of characteristic \(p\geqslant 0\), and for a point \(P\in\mathbb{P}^{2}\), let \(L_{P}\) be the Galois closure of \(k(C)/\pi^{\ast}_{P}(\mathbb{P}^{1})\), where \(k(C)\) is the function field of \(C\), and \(\pi_{P}: C \dashrightarrow \mathbb{P}^{1}\) is the projection from \(P\).
Based on the work [S. Fukasawa et al., “Algebraic curves admitting the same Galois closure for two projections”, Ann. Mat. Pura Appl (2022; doi:10.1007/s10231-022-01191-0)], and considering the characteristic different from \(2\), in this article the author provides new examples of plane curves in which the Galois closures \(L_{P_{1}}\) and \(L_{P_{2}}\) of projections from two points \(P_{1}\) and \(P_{2}\) are the same.

MSC:

14H05 Algebraic functions and function fields in algebraic geometry
14H37 Automorphisms of curves
14H50 Plane and space curves
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References:

[1] Cifani, M. G.; Cuzzucoli, A.; Moschetti, R., Monodromy of projections of hypersurfaces, Ann. Mat. Pura Appl, 201, 637-654 (2022) · Zbl 1485.14050
[2] Fukasawa, S., A birational embedding of an algebraic curve into a projective plane with two Galois points, J. Algebra, 511, 95-101 (2018) · Zbl 1400.14084
[3] Fukasawa, S.; Higashine, K.; Takahashi, T., Algebraic curves admitting the same Galois closure for two projections, Ann. Mat. Pura Appl (2022) · Zbl 1504.14052 · doi:10.1007/s10231-022-01191-0
[4] Pirola, G. P.; Schlesinger, E., Monodromy of projective curves, J. Algebraic Geom, 14, 623-642 (2005) · Zbl 1084.14011
[5] Yoshihara, H., Function field theory of plane curves by dual curves, J. Algebra, 239, 340-355 (2001) · Zbl 1064.14023
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