The essential skeleton of a product of degenerations

Morgan V. Brown, Enrica Mazzon

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration XR changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to XK. Using the Kato fan, we define a skeleton Sk(XR) when the model XR is log-regular. We show that if XR and YR are log-smooth, and at least one is semistable, then Sk(XR ×R YR) ≈ Sk(XR) × Sk(YR). The essential skeleton Sk(XK), defined by Mustaţǎ and Nicaise, is a birational invariant of XK and is independent of the choice of R-model. We extend their definition to pairs, and show that if both XK and YK admit semistable models, Sk(XK ×K YK) ≈ Sk(XK)×Sk(YK). As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the 2n-dimensional degeneration is homeomorphic to a point, n-simplex, or ℂℙn, depending on the type of the degeneration.

Original languageEnglish (US)
Pages (from-to)1259-1300
Number of pages42
JournalCompositio Mathematica
Issue number7
StatePublished - Jan 1 2019


  • Berkovich spaces
  • Kato fan
  • dual complex

ASJC Scopus subject areas

  • Algebra and Number Theory


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