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<p>We address two longstanding open problems, one originating in PL topology, another in birational geometry. We prove the weighted version of Oda’s <i>strong factorization conjecture</i> (1978): we prove that any two toric varieties whose fans have the same support admit a common toric modification induced by iterated stellar subdivisions. This implies that every two PL homeomorphic polyhedra have a common stellar subdivision, which was a conjecture going back to Tietze’s formulation of the Hauptvermutung in 1908, and often attributed to Alexander.</p>

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All triangulations have a common stellar subdivision

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Karim A. Adiprasito,
  • Igor Pak

摘要

We address two longstanding open problems, one originating in PL topology, another in birational geometry. We prove the weighted version of Oda’s strong factorization conjecture (1978): we prove that any two toric varieties whose fans have the same support admit a common toric modification induced by iterated stellar subdivisions. This implies that every two PL homeomorphic polyhedra have a common stellar subdivision, which was a conjecture going back to Tietze’s formulation of the Hauptvermutung in 1908, and often attributed to Alexander.