Related papers: A tropical view on the Bruhat-Tits building of SL …
We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,\epsilon R)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of…
We introduce and study a family of countable groups constructed from Euclidean buildings by "removing" suitably chosen subsets of chambers.
We investigate a toroidal compactification of the moduli space of K3 surfaces of degree $2$ originating from the program formulated by Gross-Hacking-Keel-Siebert. This construction uses Dolgachev's formulation of mirror symmetry and the…
We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to Bruhat-Tits buildings, for which we also provide a…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…
In this article, we study the geometric invariant theory (GIT) compactification of quintic threefolds. We study singularities, which arise in non-stable quintic threefolds, thus giving a partial description of the stable locus. We also give…
We compute the Euler-Poincar\'e characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use…
We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in R^r as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to R^r. The weights of the…
We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study…
We introduce the notion of tropicalization for Poisson structures on $\mathbb{R}^n$ with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a…
Let $X$ be a spherical variety. We show that Tevelev and Vogiannou's tropicalization map from $X$ to its tropicalization factors through the Berkovich analytification $X^{\text{an}}$, as in the case for toric varieties. Furthermore we show…
Metrical completeness for Bruhat-Tits buildings is a natural and useful condition. In this paper we determine which Bruhat-Tits buildings are metrically complete up to certain cases involving infinite-dimensionality and residue…
The spaces of point configurations on the projective line up to the action of $\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and…
We consider two sign characters defined on a tamely ramified maximal torus T of a twisted Levi subgroup M of a reductive p-adic group G. We show that their product extends to the stabilizer M(F)_x of any point x in the Bruhat-Tits building…
Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups {\Gamma}, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases.…
Let $\mathbf{G}$ be a reductive Chevalley group scheme (defined over $\mathbb{Z}$). Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve over a field $\mathbb{F}$. Let $P$ be a closed point on $\mathcal{C}$. Let $A$ be…
We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind,…
We compute the wonderful compactification of symmetric varieties of $SL(2,F)$, where $F$ is a finite field-extension of $Q_p$ with $p\neq 2$, that comes from either an abstract or $F$-involutions of $SL(2,F)$. For each of those wonderful…
The given study uses the methods to identify compactifications of semigroups $S\subset L(X),$ which reside in the space $L(X).$ This method generalizes in some sense the deLeeuw-Glicksberg-Theory to a greater class of functions. The…