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Logarithmically regular morphisms.

Sam Molcho1, Michael Temkin1

  • 1Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv'at Ram, 91904 Jerusalem, Israel.

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Summary
This summary is machine-generated.

This study introduces a combinatorial presentation for the stack of log schemes, simplifying the analysis of log morphisms. It establishes conditions for the equivalence of weak and strong properties, advancing the understanding of logarithmic regularity.

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Area of Science:

  • Algebraic Geometry
  • Number Theory
  • Commutative Algebra

Background:

  • Log schemes and log morphisms are fundamental objects in modern algebraic geometry, extending classical notions.
  • Olsson's definitions of weak and strong properties of log morphisms provide a framework for studying these objects.
  • Understanding the relationship between these properties is crucial for developing deeper insights into logarithmic structures.

Purpose of the Study:

  • To provide a concrete combinatorial presentation of the stack parametrizing log schemes.
  • To establish a criterion for when weak and strong properties of log morphisms coincide.
  • To apply these findings to the study of logarithmic regularity and develop a chart criterion for it.

Main Methods:

  • Development of a combinatorial description for the stack .
  • Analysis of weak and strong properties of log morphisms using the combinatorial presentation.
  • Application of the derived criterion to investigate logarithmic regularity and its properties.

Main Results:

  • A simplified, combinatorial presentation of the stack is established.
  • A clear criterion is proven for the equivalence of weak and strong properties of log morphisms.
  • The study derives key properties of logarithmic regularity and introduces a chart criterion analogous to Kato's.

Conclusions:

  • The combinatorial approach simplifies the study of log schemes and morphisms.
  • The established criterion unifies the understanding of different properties of log morphisms.
  • The results provide new tools for investigating logarithmic regularity in algebraic geometry.