Faltings modular height and self-intersection of dualizing sheaf

Details Faltings modular height and self-intersection of dualizing sheaf Faltings modular height and self-intersection of dualizing sheaf

1994-02-17

arxiv.org/abs/alg-geom/9402013v1

Mathematics

Algebraic Geometry

10 pages, AmSTeX

Scientific paper

Let K be a number field, O_K the ring of integers of K and X a stable curve over O_K of genus g >= 2. In this note, we will prove a strict inequality ( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)), where $K_{X/S}$ is the canonically metrized dualizing sheaf of X over S = Spec(O_K) and Height_{Fal}(J(X_K)) is the Faltings modular height of the Jacobian of X_K. As corollary, for any constant A, the set of all stable curves X over O_K with ( (K_{X/S})^2 / [K : Q] ) <= A is finite under the following equivalence. For stable curves X and Y, X is equivalent to Y if X is isomorphic to Y over O_{K'} for some finite extension field K' of K.

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