Chapter 4 - The dual of the Kummer

Summary

Introduction. A reader primarily interested in the computational aspects may omit this chapter and the next at a first reading.

Classically, over an algebraically closed field, the Kummer surface is isomorphic to its projective space dual. This is no longer the case in our situation. In this chapter we construct the dual. We do so not directly, but by solving an apparently quite different problem.

The jacobian of a curve of genus g was first constructed not from Pic^g but from Picⁿ for some n > g. Here we first obtain a variety which parametrizes Pic³ modulo the involution induced by the involution ±Y of the curve C. It turns out to be a quartic surface K* in ℙ³. A direct, if somewhat mysterious, computation shows that K* is indeed the required dual. The duality is described in terms of the canonical map from Pic² ⊕ C to Pic³ which maps the (class of the) divisor {a, b} and the point c into the class of the divisor {a, b, c}.

The chapter ends with a number of miscellaneous applications.

Description of Pic³. We use a construction of Jacobi (1846), see also [Mumford (1983), II, p. 317]. A divisor class defined over the ground field (k, as usual) does not necessarily contain a divisor defined over k. Hence we work over the algebraic closure, though the final formulae will be in k.