Constructing smooth and fully faithful tropicalizations for Mumford curves

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: fully faithful tropicalizations, which are embeddings such that the tropicalization admits a continuous section to the associated Berkovich space $$X^{{{\,\mathrm{an}\,}}}$$ X an of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.

On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups

In this paper, we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of [Formula: see text] into [Formula: see text] for [Formula: see text], where [Formula: see text] is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self-product of theta divisor into the self-product of the Jacobian of a smooth projective curve.

ON WłODARCZYK'S EMBEDDING THEOREM

We prove the following version of Włodarczyk's Embedding Theorem: Every normal complex algebraic [Formula: see text]-variety Y admits an equivariant closed embedding into a toric prevariety X on which [Formula: see text] acts as a one-parameter-subgroup of the big torus T⊂X. If Y is ℚ-factorial, then X may be chosen to be simplicial and of affine intersection.

Free finitary algebras on compactly generated spaces

An explicit colimit formula is used to describe the free k-space algebra on a given k-space for any k-enriched finitary theory. A question, raised and solved affirmatively by several authors, has been that of whether the free k-space group on a weakly hausdorff k-space is again weakly hausdorff and admits a closed embedding of the generators. In the present article both these features of finitary k-space algebra are combined to answer analogous questiona regarding the free finitary k-space algebras in general, and the weakly hausdorff separation axiom. Relationships with other problems in k-space theory are described.

Constructing Locally Flat Embeddings of Infinite Dimensional Manifolds without Tubular Neighborhoods

All spaces in this paper will be separable and metric. A closed embedding i: M → TV is said to be locally flat (of codimension n) if for each x0 ∈ M there is an open set U in M containing xo and an open embedding h: U X Rn → N for which h(x, 0) = i(x), for all x ∈ U.