Tropical compactification in log-regular varieties

In this article we define a natural tropicalization procedure for closed subsets of log-regular varieties in the case of constant coefficients and study its basic properties. This framework allows us to generalize some of Tevelev's results on tropical compactification as well as Hacking's result on the cohomology of the link of a tropical variety to log-regular varieties.Comment: 15 pages; fixed an index mistake in Thm. 1.4 and Cor. 1.5 and improved the exposition; to appear in Math. Zeitschrif

Functorial tropicalization of logarithmic schemes: The case of constant coefficients

The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of K. Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.Comment: v4: 33 pages. Restructured introduction, otherwise minor changes. To appear in the Proceedings of the LM

Tropical geometry of moduli spaces of weighted stable curves

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces.Comment: 25 pages, minor revisions, added further pictures as well as Section 6.2 discussing a tropical analogue of Kapranov's construction of \Mbar_{0,n}. To appear in the Journal of the London Mathematical Societ

Skeletons of Prym varieties and Brill--Noether theory

We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym-Brill-Noether locus for generic unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym-Brill-Noether Theorem for generic unramified double covers that is originally due to Welters and Bertram.Comment: 32 pages, 7 figures. Accepted for publication in Algebra & Number Theor

Logarithmic Picard groups, chip firing, and the combinatorial rank

Illusie has suggested that one should think of the classifying group of M_X^{gp}-torsors on a logarithmically smooth curve $X$ over a standard logarithmic point as a logarithmic analogue of the Picard group of $X$. This logarithmic Picard group arises naturally as a quotient of the algebraic Picard group by lifts of the chip firing relations of the associated dual graph. We connect this perspective to Baker and Norine’s theory of ranks of divisors on a finite graph, and to Amini and Baker’s metrized complexes of curves. Moreover, we propose a definition of a combinatorial rank for line bundles on $X$ and prove that an analogue of the Riemann–Roch formula holds for our combinatorial rank. Our proof proceeds by carefully describing the relationship between the logarithmic Picard group on a logarithmic curve and the Picard group of the associated metrized complex. This approach suggests a natural categorical framework for metrized complexes, namely the category of logarithmic curves

Altered Chromatin Occupancy of Master Regulators Underlies Evolutionary Divergence in the Transcriptional Landscape of Erythroid Differentiation

Erythropoiesis is one of the best understood examples of cellular differentiation. Morphologically, erythroid differentiation proceeds in a nearly identical fashion between humans and mice, but recent evidence has shown that networks of gene expression governing this process are divergent between species. We undertook a systematic comparative analysis of six histone modifications and four transcriptional master regulators in primary proerythroblasts and erythroid cell lines to better understand the underlying basis of these transcriptional differences. Our analyses suggest that while chromatin structure across orthologous promoters is strongly conserved, subtle differences are associated with transcriptional divergence between species. Many transcription factor (TF) occupancy sites were poorly conserved across species (∼25% for GATA1, TAL1, and NFE2) but were more conserved between proerythroblasts and cell lines derived from the same species. We found that certain cis-regulatory modules co-occupied by GATA1, TAL1, and KLF1 are under strict evolutionary constraint and localize to genes necessary for erythroid cell identity. More generally, we show that conserved TF occupancy sites are indicative of active regulatory regions and strong gene expression that is sustained during maturation. Our results suggest that evolutionary turnover of TF binding sites associates with changes in the underlying chromatin structure, driving transcriptional divergence. We provide examples of how this framework can be applied to understand epigenomic variation in specific regulatory regions, such as the β-globin gene locus. Our findings have important implications for understanding epigenomic changes that mediate variation in cellular differentiation across species, while also providing a valuable resource for studies of hematopoiesis