Abstract: This work is part of a broader program to develop necessary commutative algebra tools for the semirings arising from tropicalization. We will discuss different notions of integrality which while equivalent for rings are not for idempotent semirings. We will define integral closure and give different characterizations. If time permits we will give some examples of normalization in tropical geometry. 

Abstract: We study asymptotic behavior of the stability thresholds of a big line bundle, and prove explicit bounds on the error terms. This improves a theorem of Jin--Rubinstein--Tian by removing the assumption on the existence of a divisorial minimizer. A key step in our proof is to show that the stability thresholds of a big line bundle can always be computed by quasi-monomial valuations. This generalizes Blum--Jonsson's result on the stability thresholds of an ample line bundle.

Abstract: Semirings naturally arise in various contexts, including tropical geometry and geometry over the field with one element. In this talk, I will introduce some basic aspects of semirings and explore their applications to scheme theory over semirings, with a particular focus on line bundles and vector bundles. This is joint work with James Borger.

Abstract: A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed numerical invariants, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for some line bundles related to the determinant line bundle on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.

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