• Tags
  • #todo/untagged
• Refs:
  • Mattia Talpo: Kummer-étale additive invariants of log schemes. https://www.youtube.com/watch?v=r-CXf24pE9Y
• Links:
  • #todo/create-links

Log geometry

# Notes

Reference: Mattia Talpo: Kummer-étale additive invariants of log schemes. https://www.youtube.com/watch?v=r-CXf24pE9Y

• Origins: Fontaine-Illusie-Kato (Deligne, Faltings) in the late 80s, schemes/stacks/derived schemes plus additional derived (“log”) structure.

• Examples:

  • A pair \((X, D)\) of a smooth divisor.
  • Or \(D\) a toroidal embedding, remembers the boundary \(D\)
    • toric boundary
  • Think about this like "“\(X\) with a marked point on the boundary”

• How they arise: in characteristic zero, working with a non-compact scheme. Compactify, and use divisor.

• Every log scheme has a locus where the log structure is trivial, \(\left\{{x\in X{~\mathrel{\Big\vert}~}{\mathcal{O}}_{X, x}^{\times}= M_x}\right\}\), so stalk of a sheaf of \(M\) are units. Think of this like the complement of the boundary.

  • Trivial part of a log scheme pair \((X, D)\) is the complement of the divisor.
    • Rank of the free monoid records number of branches passing through singular points:

  

• Logarithmic geometry generalizes toroidal geometry to non-smooth settings.

• See semistable degeneration