By Ingrid Bauer, Shelly Garion, Alina Vdovina

This selection of surveys and examine articles explores a desirable classification of types: Beauville surfaces. it's the first time that those items are mentioned from the issues of view of algebraic geometry in addition to workforce concept. The publication additionally comprises numerous open difficulties and conjectures on the topic of those surfaces.

Beauville surfaces are a category of inflexible average surfaces of common kind, which might be defined in a merely algebraic combinatoric means. They play a massive function in numerous fields of arithmetic like algebraic geometry, staff thought and quantity idea. The concept of Beauville floor was once brought by way of Fabrizio Catanese in 2000 and after the 1st systematic learn of those surfaces by means of Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald, there was an expanding curiosity within the subject.

These court cases mirror the subjects 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 other fields of arithmetic attracted to Beauville surfaces.

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**Example text**

There are remarkable connections between the group G K and the group G 0 discussed in Sect. 1. Firstly, we can present the group G K in an alternative way with different generators: G K = w0 , . . ,w6 , y0 , . . , y6 , z 0 , . . , z 6 | wi wi+1 wi+3 , yi yi+1 yi+3 , z i z i+1 z i+3 , wi−1 y6(1+i) z i−1 , wi−1 z i−1 y6(1+i) (i ∈ Z7 ) , (5) where each of the three subsets of generators has very similar relators like those appearing for the group G 0 in (1), with only two more series of relators added representing connections between the generators of different subsets.

1 below) to create infinitely many groups with special presentations. More precisely, he constructed a special presentation with star graph isomorphic to the incidence graph of the projective plane over every finite field Fq , where q is a prime power. , the incidence graph of the 7-point projective plane over F2 ). Given a finite field K = Fq (for q a prime power), a positive presentation with star graph isomorphic to this incidence graph of the Desarguesian projective plane over K is formed. The construction takes a cubic extension of K , namely F = Fq 3 , and identifies the cyclic group Cm = F × /K × with the points of the projective pane P over K , where m = q 2 + q + 1.

Eq : Zq−1 or 2 B2 (q). (b) If x, √ y ∈ 2 B2 (q) are two elements such that o(x) and o(y) are have orders dividing q ± 2q + 1 and o(xy) = 2, then x, y = 2 B2 (q) or a subfield subgroup. Proof of Theorem 9 For our first generating pair we consider the following elements of 2 B2 (q) each of which are easily checked to have order 2 by direct calculation. Strongly Real Beauville Groups ⎛ 0 ⎜0 t1 := ⎜ ⎝0 1 0 0 1 0 0 1 0 0 47 ⎞ 0 0 0 β −1 n+1 ⎜0 0 β −2 +1 0 ⎟ ⎟ t2 := ⎜ n+1 ⎝ 0 β 2 −1 0 0 ⎠ β 0 0 0 ⎞ 1 0⎟ ⎟ 0⎠ 0 ⎛ ⎛ 1 0 ⎜ 0 1 t3 := ⎜ ⎝ α2n+1 0 n+1 α2 α2 0 0 1 0 ⎞ 0 0⎟ ⎟ 0⎠ 1 where α and β are generators of the multiplicative group F× q .