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Non-homeomorphic Galois conjugate Beauville structures on \(\mathrm{PSL}(2,p)\). (English) Zbl 1263.14038

The authors construct arbitrarily large families of Galois conjugate Beauville surfaces with the property that they are pairwise non-homeomorphic. As usual, the construction starts with the action of some finite group \(G\) acting on two triangle curves \(C_1,C_2\) such that the canonical projections \(C_i \to C_i/G\) are Belyi functions (in dessins theory, such curves are often called “quasiplatonic”). If the diagonal action of \(G\) on \(C_1 \times C_2\) is fixed point free, the resulting compact complex surface \(B := (C_1 \times C_2)/G\) is a Beauville surface – and has very interesting properties.
In the present paper, the special case \(G = \text{PSL}(2,p)\) is considered and the curves are considered in such a way that \(C_1\) is uniquely determined by the action of \(G\) and its signature – hence \(C_1\) is defined over the rationals – whereas \(C_2\) admits a proper Galois action, inducing a proper Galois action on the Beauville surface. A striking result of Catanese implies that the Beauville surfaces of this Galois orbit have non-isomorphic fundamental groups, hence are not homeomorphic. The lexicographic smallest example arises for \(p=7\) and genera \(8\) abd \(49\) for the curves \(C_i\).
On the way to this result, the authors give a very well readable introduction to quasiplatonic curves and all techniques necessary for the construction of Beauville surfaces, among others a new and relatively elementary proof of Catanese’s rigidity theorem.

MSC:

14J25 Special surfaces
14H30 Coverings of curves, fundamental group
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
32J15 Compact complex surfaces
30F10 Compact Riemann surfaces and uniformization
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