311 research outputs found
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 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 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 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 ERRATUM: This paper is withdrawn.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 from Conics
We study GIT quotients parametrizing n-pointed conics that generalize the GIT
quotients . Our main result is that
admits a morphism to each such GIT quotient, analogous to the well-known result
of Kapranov for the simpler 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 ordered points in
and of points contained in a union of linear subspaces . 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
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
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