Self-intersection of dualizing sheaves of arithmetic surfaces with reducible fibers
Abstract
Let K be an algebraic number field and OK the ring of integers of K. Let f : X --> Spec(OK) be a stable arithmetic surface over OK of genus g >= 2. In this short note, we will prove that if f has a reducible geometric fiber, then the self intersection of dualizing sheaf of X with Arakelov metric is greater than or equal to log(2)/6(g-1).
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