Effective Faithful Tropicalizations Associated to Linear Systems on Curves

Released on by Shu Kawaguchi
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  • Effective Faithful Tropicalizations Associated to Linear Systems on Curves Book Detail

  • Author : Shu Kawaguchi
  • Release Date : 2021-07-21
  • Publisher : American Mathematical Soc.
  • Genre : Education
  • Pages : 110
  • ISBN 13 : 1470447533
  • File Size : 11,11 MB
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Effective Faithful Tropicalizations Associated to Linear Systems on Curves by Shu Kawaguchi PDF Summary

Book Description: For a connected smooth projective curve X of genus g, global sections of any line bundle L with deg(L) ≥ 2g + 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since X is a curve, an isometry), which is called a faithful tropicalization. Let K be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that X is defined over K and has genus g ≥ 2 and that Γ is a skeleton (that is allowed to have ends) of the analytification Xan of X in the sense of Berkovich. We show that if deg(L) ≥ 3g − 1, then global sections of L give a faithful tropicalization of Γ into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree.

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