On Beauville surfaces

Abstract

Abstract. We prove that if a finite group G acts freely on a product of two curves C 1 C 2 so that the quotient S D C 1 C 2 =G is a Beauville surface then C 1 and C 2 are both non hyperelliptic curves of genus 6; the lowest bound being achieved when C 1 D C 2 is the Fermat curve of genus 6 and G D .Z=5Z/ 2 . We also determine the possible values of the genera of C 1 and C 2 when G equals S 5 , PSL 2 .F 7 / or any abelian group. Finally, we produce examples of Beauville surfaces in which G is a p-group with p D 2; 3.Mathematics Subject Classification (2010). 14J29, 20D06, 30F10.

New examples of Beauville surfaces

“…Hence, the only p-groups that can admit a mixed Beauville structure are 2-groups. However in the unmixed case, by [2] and [12], for every prime number p there exists a p-group admitting an unmixed Beauville structure.…”

On Beauville structures on the groups S n and A n

“…In the next example we employ this theorem to produce an unmixed Beauville structure on A 6 . It has been already noted that A n does not admit a Beauville structure for n ≤ 5 [2,6].…”

“…Most of what is known about these problems is due to Catanese [4] and Bauer et al [2,3]. See also our article [6].…”

On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

“…In , Fuertes, González‐Diez, and Jaikin‐Zapirain comment that it is very plausible that most 2‐generator finite ‐groups of sufficiently large order are Beauville groups. In this paper, we shed some light on this problem, by showing that there are an abundance of both Beauville and non‐Beauville groups among 2‐generator finite ‐groups.…”

“…Also, in general, the construction of Beauville ‐groups is much more difficult when or 3 than for . Actually, for some time it was not even known whether there existed any Beauville 2‐groups or 3‐groups, and the first examples were given in [, Section 5]. It is certainly an interesting problem to try to find a lower bound for when or 3, but it will require a radically different approach.…”