By Ingrid Bauer, Shelly Garion, Alina Vdovina

This number of surveys and learn articles explores a desirable category of sorts: Beauville surfaces. it's the first time that those items are mentioned from the issues of view of algebraic geometry in addition to team idea. The booklet additionally contains quite a few open difficulties and conjectures concerning those surfaces.

Beauville surfaces are a category of inflexible normal surfaces of basic style, which might be defined in a basically algebraic combinatoric method. They play an enormous function in several fields of arithmetic like algebraic geometry, workforce idea and quantity concept. The suggestion of Beauville floor was once brought by means of Fabrizio Catanese in 2000 and after the 1st systematic learn of those surfaces by way of Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald, there was an expanding curiosity within the subject.

These lawsuits mirror the themes of the lectures provided through the workshop ‘Beauville surfaces and teams 2012’, held at Newcastle college, united kingdom in June 2012. This convention introduced jointly, for the 1st time, specialists of alternative fields of arithmetic drawn to Beauville surfaces.

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**Example text**

Similarly we find that t1 t2 t1 t3 has characteristic polynomial of the correct form to have order q − 1. From Lemma 11(a) we see that these elements generate the group since x1 and y1 are not both contained in a cyclic subgroup (one of them is diagonal) and by direct calculation no one-dimensional subspace in the natural module is preserved by them so there is no proper subgroup containing each of these elements. n+1 n+1 2n+1 n+1 n+1 48 B. Fairbairn For the second triple we consider the matrices ⎛ 0 ⎜0 x2 := ⎜ ⎝0 1 0 0 1 0 0 1 0 δ4 ⎞ 1 0⎟ ⎟ δ4 ⎠ δ2 ⎛ 2 4 0 ⎜ 4 0 1 y2 := ⎜ ⎝0 1 0 1 0 0 ⎞ 1 0⎟ ⎟ 0⎠ 0 where δ, ∈ Fq are chosen so that δ = and these do not have the correct form for these elements to have order q − 1.

C. Bauer, F. Catanese, F. Grunewald, The classification of surfaces with pg = q = 0 isogenous to a product of curves. Pure Appl. Math. Q. 4(2), 547–586 (2008) 7. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system, I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997) 8. Y. Fuertes, G. González-Diez, A. Jaikin-Zapirain, On Beauville surfaces. Groups Geom. Dyn. 5(1), 107–119 (2011) 9. Y. A. Jones, Beauville surfaces and finite groups. J. Algebra 340(1), 13–27 (2011) 10. J. Howie, On the SQ-universality of T (6)-groups.

Algebra 215(11), 2780–2788 (2011) 23. A. Sarveniazi, Explicit construction of a Ramanujan (n 1 , n 2 , . . , n d−1 )-regular hypergraph. Duke Math. J. 139(1), 141–171 (2007) 24. F. Serrano, Isotrivial fibred surfaces. Ann. Mat. Pura Appl. 171(4), 63–81 (1996) A Survey of Beauville p-Groups Nigel Boston Abstract This paper describes recent results as to which p-groups are Beauville, with emphasis on ones of small order (joint with N. Barker and B. Fairbairn) and ones that form inverse systems (joint with N.