ISRAEL JOURNAL OF MATHEMATICS, Vol. 74, Nos. 2-3, 1991 BRANCHED TWO-SHEETED COVERS BY HERSHEL M. FARKASa't AND IRWIN KRAb'l" alnstitute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel; and bDepartment of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA ABSTRACT In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve in PCg-I . In particular, we consider non-hyperelliptic Riemann surfaces with invo- lution and the Weierstrass points of the - 1 eigenspace of the holomorphic differ- entials. The case of coverings of a torus is considered in detail. 1. Introduction The current paper originated as an attempt to understand the following situa- tion. If S is a non-hyperelliptic compact Riemann surface of genus 3, then the ca- nonical curve representing S is a quartic in PC 2. It is well known that such a curve has 28 bitangents; see, for example, [3]. Examples can be constructed where 4 of these bitangents pass through a common point and the 8 points of bitangency are the intersection of the curve with a homogeneous quadric. It is easy to see that any non-hyperelliptic surface S of genus 3 which is a branched two-sheeted cover p of a torus X satisfies the above condition. In this in- troduction we shall sketch the proof of this assertion. In (the remainder of) this paper we generalize this result and hence offer a (possible) explanation of what really lies behind it. If S is a non-hyperelliptic surface of genus 3 and S admits a conformal involu- tion E, then it follows that E has precisely 4 fixed points. This is a consequence of the fact that on a surface of genus 3, a conformal involution can have either 0, tResearch of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis(supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University. Received September 13, 1990 169