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By David A. Cox

Replicate symmetry started while theoretical physicists made a few mind-blowing predictions approximately rational curves on quintic hypersurfaces in 4-dimensional projective area. realizing the math in the back of those predictions has been a considerable problem. This booklet is the 1st thoroughly finished monograph on reflect symmetry, protecting the unique observations via the physicists during the newest growth made so far. topics mentioned comprise toric forms, Hodge concept, Kähler geometry, moduli of good maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the replicate theorem

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To deal with the divisor classes of degree < g — 1 we can first translate them by a divisor with positive degree and then apply the previous argument. In particular, the number of divisor classes of any degree is equal to the number of divisor classes of degree 0, which is simply the cardinality of C1O(C). This proves the lemma. Remark C1O(Q is the analogue of the class group of an algebraic number field; thus when convenient we shall refer to the cardinality of C1O(C) as the class number of the curve C and denote it by h{C).

Are linearly independent over k. We want to show that the ed elements tiUj, 1 < i < e, 1

Let k be the algebraic closure of the finite field F, and define a group H = {X e kx: kq*1 = 1}. Drienfeld has studied the action of SL2(F,) by linear change of coordinates and of H by homothety on the affine irreducible curve ^ = {(x,y)ek2:xyq~x'>y= 1}. Show how to extend this action to the non-singular projective curve Show that over the field F,2 the curve % can be written as xt+l + yq+l + zq+l on which the unitary group l/3(F,) acts naturally. What relation is there between these two group actions?

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