An Introduction to Algebraic Geometry by K. Ueno

By K. Ueno

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2) where {(f),ip) are homogeneous coordinates. A chord is a line joining two points of F. e. 3) 5(Am - /xl, Am + /il) = A 2 5(m) - »2S(l) =0-0=0 ^ where 1, m are on F. e. the quadric surface S ^ X ) = 0 has a single singularity, namely at X = x. f Consider the generators of the quadrics through F and x. 5) has 5 i (x,y) = 0 (i = 1,2,3). 6) A point x is on the Jacobian of 5i, 5 2 , S3, S 4 , in the sense of the previous section, if there is a y such that 5i(x,y) = 0 (j = 1,2,3,4). 7) As we saw above, the first three of these conditions, at least for general x, are equivalent to the existence of A, JU, 1, m such that (3), (5) hold, where without loss of (much) generality we may suppose that m = (l,«,« 2 V), for some -0, u.

The reader familiar with the theory of elliptic curves will note analogies: the analogue of the jacobian and the Kummer are the elliptic curve itself (Y2 = X 3 -f AX + B, say) and the line with coordinate X. In this chapter we describe the Kummer and some of its more basic properties. These are used to give an explicit treatment of the group law on the jacobian. In a final section we discuss when a surface with the geometric properties of a Kummer surface actually arises from a curve. We do not reproduce some of the more complicated formulae.

34 Chapter 4. The dual of the Kummer The argument so far is reversible. Given a point of /C* with 7/4 ^ 0, we may take 7/4 = 1 by homogeneity. The quadratic form YlVjSj i s singular, and so^f is represent able as W2 — [//Vj. The transformation Xj H+ X^ now gives an effective divisor of degree 3 on C. Further, the class of the divisor up to linear equivalence and the ±Y involution depends only on the rjj. Hence the corresponding element of Pic3 is either defined over k or defined over a quadratic extension, its conjugate over k being its conjugate under the ±Y involution.

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