Convex Polytopes, Algebraic Geometry, and Combinatorics

  title={Convex Polytopes, Algebraic Geometry, and Combinatorics},
  author={Laura Escobar and Kiumars Kaveh},
  journal={Notices of the American Mathematical Society},
In the last several decades, convex geometry methods have proven very useful in algebraic geometry specifically to understand discrete invariants of algebraic varieties. An approach to study algebraic varieties is to assign to a family of varieties a corresponding family of combinatorial objects which encode geometric information about the varieties. Often, the combinatorial objects that arise are convex polytopes, and convex geometry has been an essential tool for this strategy. The emergence… 
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  • F. M
  • Mathematics
  • 1983
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