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By Robin Hartshorne, C. Musili

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In particular, every curve lies over P1 . Clearly, if C ⇒ C and C ⇒ C , then C ⇒ C . We say that a curve C is minimal for some class of curves C if every C ∈ C lies over C . Let Cn : y n = x2 + 1 (1) and C be the set of such curves. For all n, m ∈ N we have the standard, ramified, map Cmn → Cn , y → y m . At the same time, Cmn ⇒ Cn . Belyi’s theorem [1] implies that for every curve C defined over a number field there exists a curve C = Cn ∈ C such that C ⇒ C (see [4] for a simple proof of this corollary).

Let σ ∈ U be a double-transposition. We have m(σ) = 4. Let m now be the minimal degree taken over all nontrivial elements of U . Since U is Beauville surfaces without real structures 29 also primitive we may apply a result of de S´eguier (see [15], page 43) which says that if m > 3 (in our situation we would have m=4) then n< m m 3 m2 log + m log + 4 2 2 2 . 5. Our assumptions imply that m = 3. We then apply Jordan’s theorem to reach the desired conclusion. We now construct the systems of generators required for the constructions of Beauville surfaces.

9. Let X(7) : x3 y + y 3 z + z 3 x = 0 be Klein’s quartic plane curve of genus 3. Using (x, y, z) → (z 3 /x2 y, −z/x) we see that X(7) is isomorphic to the curve y 7 = x2 (x + 1) while C7 is isomorphic to y 7 = x(x + 1). Thus their fiber product over P1 is unramified for both projections so that X(7) ⇔ C7 . 50 Fedor Bogomolov and Yuri Tschinkel 3 A graph on the set of elliptic curves Axiomatizing the constructions of Section 2, we are lead to consider a certain ¯ directed graph structure on the set E of all elliptic curves defined over Q, defined as follows: Write E E , resp.

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