38 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
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
Equations of tropical varieties
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as =(ℝ∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of -points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of -schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting