The Equations Defining a Curve of Genus 41

For most curves of genus 4 and characteristic > 3 the second osculating cone of the theta divisor is the cone over the canonical curve. Let G be a nonhyperelliptic complete smooth curve of genus 4. The canonical morphism G —> P3 is an embedding and the image is defined as the intersection of a quadric and a cubic. In this paper we will discuss a geometric way of constructing such equations from the theta divisor on the Jacobian of C. Recall that the singularities of the theta divisor correspond to linear systems \D\ on C of dimension one and degree 3. For a general curve there are two such systems |D| and \K — D\. for some special curves there is only one \D\ — \K — D\. We will give a construction of the equations of the canonical curve when the theta divisor has two singular points. This construction was suggested by the work of Ehrenpreis and Farkas [1] where they show that it works for an unknown but sufficiently general Riemann surface. We will indicate the construction in the Riemann surface case. Let C4 be the universal covering space of the Jacobian. Let x in C4 be a point corresponding to a g\, say \D\, on C. We make the power series expansion 6(y) = /j + /j ■)-of the theta function at x where /¿ is a homogeneous polynomial of degree i in y x. The polynomials f2 and f$ may be identified with forms on the canonical space P3. As such their zeroes contain the canonical curve. We can prove that if |2D| ^ \K\, f2 and fo define the canonical curve. (If \2D\ = K, f3 = 0 and the method fails.) We will formulate our results algebraically so that they hold for a curve over any field of characteristic not equal to two. 1. Flat varieties and osculating cones. We will be working with schemes of finite type over an algebraically closed field k. Points will always be fc-rational. If z is a point on a scheme X, for any positive integer n the scheme Xnx is the nth infinitesimal neighborhood of x. It is the closed subscheme of X defined by the power ideal Jn+1 where I is the ideal of x. A flat structure of nth order on a smooth variety X at the point x is an isomorphism i: Tno —» Xno where T is a vector variety. In this situation T may be identified with the tangent space of X at x. We henceforth will fix such a flat structure. A smooth subvariety Y of X passing through x is flat to the nth order if i~1(Ynx) has the form SnQ where S is a linear subspace of T. Here S may be identified with the tangent space of Y at x. Let X\,... ,Xn be regular functions on X at x such that i*X,... ,i*Xn are a basis for the linear functions on T„oThe Xj are determined modulo Jn+1 by their Received by the editors October 8, 1984 and, in revised form, June 25, 1985. 1980 Mathematics Subject Classification. Primary 14H40; Secondary 30F10. Partly supported by NSF grant #75-05578. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page