Yolanda Fuertes

On Beauville surfaces

We prove that if a finite group G acts freely on a product of two curves C1×C2 so that the quotient S = C1×C2/G is a Beauville surface then C1 and C2 are both non hyperelliptic curves of genus ≥ 6; the lowest bound being achieved when C1 = C2 is the Fermat curve of genus 6 and G = (Z/5Z). We also determine the possible values of the genera of C1 and C2 when G equals S5, PSL2(F7) or any abelian group. Finally, we produce examples of Beauville surfaces in which G is a p-group with p = 2, 3.

ON THE NUMBER OF COINCIDENCES OF MORPHISMS BETWEEN CLOSED RIEMANN SURFACES

We give a bound for the number of coincidence of two morphisms between given compact Riemann surfaces (complete complex algebraic curves). Our results generalize well known facts about the number of fixed points of an automorphism.

On the Lefschetz Number of Quasiconformal Self-Mappings of Compact Riemann Surfaces

A well-known theorem of Hurwitz states that if τ[ratio ] S → S is a conformal self-mapping of a compact Riemann surface of genus g [ges ]2, then it has at most 2 g +2 fixed points and that equality occurs if and only if τ is a hyperelliptic involution. In this paper we consider this problem for a K -quasiconformal self-mapping f [ratio ] S → S . The result we obtain is that the number of fixed points (suitably counted) is bounded by 2+ g ( K 1/2 + K −1/2 ), and that this bound is sharp. We see that when K =1, that is, when f is conformal, our result agrees with the classical one.