TY - JOUR
T1 - The essential skeleton of a product of degenerations
AU - Brown, Morgan V.
AU - Mazzon, Enrica
N1 - Funding Information:
authors in contact. Mazzon is grateful to the University of Miami for hosting her visit during the completion of this project. Brown was supported by the Simons Foundation Collaboration Grant 524003. Mazzon was partially supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council (PI: Johannes Nicaise), and by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
Publisher Copyright:
© Foundation Compositio Mathematica 2019.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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.
AB - 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.
KW - Berkovich spaces
KW - Kato fan
KW - dual complex
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U2 - 10.1112/S0010437X19007346
DO - 10.1112/S0010437X19007346
M3 - Article
AN - SCOPUS:85080910113
VL - 155
SP - 1259
EP - 1300
JO - Compositio Mathematica
JF - Compositio Mathematica
SN - 0010-437X
IS - 7
ER -