Extending results of Bauer, Catanese and Grunewald, and of Fuertes and González-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L 2 (q) and SL 2 (q) for all prime powers q > 5, and the Suzuki groups Sz(2 e ) and the Ree groups R(3 e ) for all odd e 3. We also show that L 2 (q) and SL 2 (q) admit strongly real Beauville structures, yielding real Beauville surfaces, for all q > 5.

It is known that the symmetric group S n , for n ≥ 5, and the alternating group A n , for large n, admit a Beauville structure. In this paper we prove that A n admits a Beauville (resp. strongly real Beauville) structure if and only if n ≥ 6 (resp n ≥ 7). We also show that S n admits a strongly real Beauville structure for n ≥ 5.

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.

Abstract. In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

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