# The essential skeleton of a product of degenerations

@article{Brown2019TheES, title={The essential skeleton of a product of degenerations}, author={Morgan V. Brown and Enrica Mazzon}, journal={Compositio Mathematica}, year={2019}, volume={155}, pages={1259 - 1300} }

We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration $\mathscr{X}_{R}$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_{K}$ . Using the Kato fan, we define a skeleton $\text{Sk}(\mathscr{X}_{R})$ when the model $\mathscr{X}_{R}$ is log-regular. We show that if $\mathscr{X}_{R}$ and $\mathscr{Y}_{R}$ are log-smooth, and at least one is semistable, then $\text{Sk}(\mathscr{X}_{R… Expand

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#### References

SHOWING 1-10 OF 74 REFERENCES

The geometry of degenerations of Hilbert schemes of points

- Mathematics
- 2018

Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional… Expand

Skeletons and tropicalizations

- Mathematics
- 2014

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model… Expand

Non-Archimedean geometry of Artin fans

- Mathematics
- 2016

The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of… Expand

The essential skeleton of a degeneration of algebraic varieties

- Mathematics
- 2013

In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let $k$ be a field of characteristic zero and let $X$ be a smooth and projective… Expand

A relative Hilbert–Mumford criterion

- Mathematics
- 2015

We generalize the classical Hilbert–Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an… Expand

Remarks on degenerations of hyper-K\"ahler manifolds

- Mathematics
- 2017

Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the… Expand

Desingularized moduli spaces of sheaves on a K3, I

- Mathematics
- 1997

Moduli spaces of semistable torsion-free sheaves on a K3 surface $X$ are often holomorphic symplectic varieties, deformation equivalent to a Hilbert scheme parametrizing zero-dimensional subschemes… Expand

The dual complex of Calabi–Yau pairs

- Mathematics
- 2015

A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that $$K_X+D$$KX+D is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic to… Expand

Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton

- Mathematics
- 2012

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a… Expand

Moduli spaces $M_{g,n}(W)$ for surfaces

- Mathematics
- 1994

We construct and prove the projectiveness of the moduli spaces which are natural generalizations to the case of surfaces of the following:
1) $M_{g,n}$, the moduli space of $n$-marked stable curves,… Expand