Transcanonical Embeddings of Hyperelliptic Curves

Throughout this paper, a curve will be an irreducible nonsingular onedimensional projective variety over an algebraically closed field F of characteristic not 2. In Chapter 1 of his beautiful survey [7], Mumford undertakes to exhibit “every curve”. The nonhyperelliptic curves are described by the equations of their canonical embeddings [8, 91, but the hyperelliptic curves are given only as ramified double coverings of Pi. In this note we describe the equations and geometry of the “transcanonical” embeddings of hyperelliptic curves, completing, in a sense, the above exhibition. I would like to thank Sebastian Xambb, who contributed a number of ideas to the formulations below. We fix for the remainder of the paper a hyperelliptic curve C of genus gr2, and let Ko be the divisor corresponding to the unique map 0 of degree 2 from C to the projective line P’ [5, IV 5.31. From the Riemann-Roth formula and the criterion [j, IV 3.11 one sees easily that the complete linear series l(g+ k)Kd is very ample iff kz 1. We call the divisor (g-t k)Ko, for kz 1, the kth transcanonical divisor (since the canonical divisor is (gl)Ko), and we write CL, for the image of C under the corresponding embedding in projective space; thus Ck is a nondegenerate (that is, not contained in a hyperplane) curve of degree 2g + 2k in P’+ 2k. We will be particularly concerned with the case k= 1, which is the simplest. We call (g+ 1)Ko the hypercanonical divisor, and the corresponding embedding CCI C WC 2 the hypercanonical embedding. As we shall see, the hypercanonical divisor could have been defined as the ramification divisor of the canonical map, and thus is well defined even if the base field is not algebraically closed, or even more generally. See [lo] for considerations of this type.