The universal tropicalization and the Berkovich analytification

Given an integral scheme X over a non-archimedean valued field k, we construct a universal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by earlier work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification of X. Moreover, using the scheme-theoretic tropicalization, we obtain a tropical scheme $Trop_{univ}(X)$ whose T-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X is affine, we show that $Trop_{univ}(X)$ is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $Trop_{univ}(X)$ represents the moduli functor of valuations on X, and when X = spec A is affine there is a universal valuation on A taking values in the semiring of regular functions on the universal tropicalization.Comment: 16 pages, added material on the Berkovich topolog

The diffeomorphism group of a K3 surface and Nielsen realization

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.Comment: 20 pages, published version. ERRATUM: This paper is withdrawn. As pointed out by Bena Tshishiku, Borel's theorem on the cohomology stable range for arithmetic groups is incorrectly quoted and applied in this paper; in the case relevant to K3 surfaces the range is unfortunately zero. Hence the proof of the part of Theorem 1.1 referring to K3 surfaces is fundamentally broken. See arXiv:1711.0313

GIT Compactifications of M0,nM_{0,n} from Conics

We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.Comment: 15 pages, 5 figures; corrected inequality in Lemma 5.1, Int. Math. Res. Notices Vol. 201

Factorization of point configurations, cyclic covers and conformal blocks

We describe a relation between the invariants of $n$ ordered points in $P^d$ and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2} \subset P^d$. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cover with weights as in this attaching map. We find that both GIT polarizations and the Hodge class for families of cyclic covers yield line bundles on $\bar{M}_{0,n}$ with functorial restriction to the boundary. We introduce a notion of divisorial factorization, abstracting an axiom from rational conformal field theory, to encode this property and show that it determines the isomorphism class of these line bundles. As an application, we obtain a unified, geometric proof of two recent results on conformal block bundles, one by Fedorchuk and one by Gibney and the second author.Comment: 17 pages, 3 figure