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Weierstrass points on arithmetic surfaces. (English) Zbl 0723.14019
Let C be a smooth, geometrically irreducible, curve of genus at \(least\quad 2,\) defined over a number field K. For K large enough, the regular minimal model \({\mathfrak X}\) of C over Spec(\({\mathfrak O}_ K)\) has semistable reduction and the ratio \(e(C)=(\omega \cdot \omega)/[K:{\mathbb{Q}}]\) is then independent of the choice of K, where \(\omega\) is the relative dualizing sheaf of \({\mathfrak X}\) over Spec(\({\mathfrak O}_ K)\) and (\(\omega\cdot \omega)\) is an Arakelov intersection product. By generalizing a method of Arakelov in the function field situation, we obtain a proof of the strict positivity of e(C) if the stable model has at least one reducible fiber or if the set of places of completely supersingular reduction is infinite. The method uses Weierstrass points and leads also to a proof of the boundedness of the average height of Weierstrass points of powers of a given line bundle on C.
Reviewer: J.-F.Burnol

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H55 Riemann surfaces; Weierstrass points; gap sequences
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References:
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