By Ingrid Bauer, Shelly Garion, Alina Vdovina

This number of surveys and study articles explores a desirable category 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 team idea. The publication additionally comprises a number of open difficulties and conjectures with regards to those surfaces.

Beauville surfaces are a category of inflexible typical surfaces of common sort, which might be defined in a only algebraic combinatoric means. They play a massive function in numerous fields of arithmetic like algebraic geometry, team concept and quantity concept. The proposal of Beauville floor used to be brought by means of Fabrizio Catanese in 2000 and after the 1st systematic learn of those surfaces through Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald, there was an expanding curiosity within the subject.

These complaints replicate the themes of the lectures offered throughout the workshop ‘Beauville surfaces and teams 2012’, held at Newcastle collage, united kingdom in June 2012. This convention introduced jointly, for the 1st time, specialists of other fields of arithmetic drawn to Beauville surfaces.

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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) Regular Algebraic Surfaces, Ramification Structures and Projective Planes 33 6. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997) 7. H. Bruck, Quadratic extensions of cyclic planes, in Proceedings Symposium in Applied Mathematics, pp. 15–44 (1960) 8. I. M. Mantero, T.

The following array illustrates a basic projection λ: 0:1 1:3 2:4 3:0 4:1 5:0 6:0 4 2 3 4 5 2 1 2 5 6 5 6 6 3. Here, every point k represents a row and is followed by the points contained in the associated line λ(k). For example, the line λ(3) consists of the points 0, 4, 5. e. for (0, 1, 3) ∈ T we also have (1, 3, 0), (3, 0, 1) ∈ T . The associated group presentation G T agrees with the presentation of G 0 in (1). 9 [18, Example 2] The projective plane P of order 4 can be partitioned by three projective planes of order two (see [7]).

J. Algebra 340(1), 13–27 (2011) 10. J. Howie, On the SQ-universality of T (6)-groups. Forum Math. 1(3), 251–272 (1989) 11. A. O’Brien, The p-group generation algorithm. J. Symb. Comput. 9, 677–698 (1990) Strongly Real Beauville Groups Ben Fairbairn Abstract A strongly real Beauville group is a Beauville group that defines a real Beauville surface. Here we discuss efforts to find examples of these groups, emphasising on the one extreme finite simple groups and on the other abelian and nilpotent groups.