×. , U. , and U. , relative effective Cartier divisor of primitive -torsion points. Since X ,U is a J(X) U -torsor over U , there is an induced map : J(X) U,prim

, Hence M is isomorphic to an irreducible component of J(X) U,prim [ ] × M ,U and via its first projection, it is surjective over J(X) U,prim, The closure of the image in X is a curve of X which intersects M infinitely many times by the non-injectivity of ?

, Shioda's conjecture holds, namely supersingular K3 surfaces are unirational. The proof we give here does not use this powerful theorem

, Let X a K3 surface over an algebraically closed field k. Suppose that the automorphism group of X is infinite. Then X contains infinitely many rational curves

S. Let and .. {c-1, Let g be an element in the kernel of ?. Then g fixes all the rational curves. By 4.2.2, every non-trivial effective divisor is linearly equivalent to a sum of rational curves. Hence g acts trivially on Pic(X). It follows that the kernel of ? is contained in the kernel of the morphism Aut(X) ? O(Pic(X)). The latter is finite by [64, Chapter 15, C r } be the finite set of rational curves on X. Then Aut(X) acts on S and defines a homomorphism ?

, Let X be a K3 surface with Picard number ? 20. Then X has infinitely many rational curves

, ] the automorphism group of X is infinite and the result follows from Proposition 4.4.1. If the Picard rank of X is equal to 20, then the height of X is finite and by [64

, is an isomorphism and K is the fraction field of W (k). Moreover, since X is not birationally ruled

(. Aut, By [99], the automorphism group of X K is infinite. Hence Aut(X) is infinite and by proposition 4.4.1 X has infinitely many rational curves

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