The basic locus of regular ramified unitary Rapoport-Zink spaces at vertex-stabilizer level
Ioannis Zachos, Zhihao Zhao
TL;DR
The work provides a concrete description of the reduced basic locus of regular ramified unitary Rapoport–Zink spaces at maximal vertex level by constructing a Bruhat–Tits stratification and introducing strata models that are étale-locally isomorphic to each BT stratum. It shows that each BT stratum can be realized as a strict transform via a blow-up of the local model at the worst point or via explicit linear-algebraic moduli data, yielding smoothness, explicit dimension formulas, and irreducibility. The main results give a precise decomposition of the reduced basic locus into two families of strata indexed by vertex lattices of type above or below the critical value, with precise intersection behavior and irreducibility preserved under blow-up. Overall, the paper advances the understanding of integral models and their BT stratifications, with potential applications to arithmetic intersection problems and related conjectures in Kudla–Rapoport and beyond.
Abstract
We construct the Bruhat-Tits stratification of the reduced basic locus of regular ramified unitary Rapoport-Zink spaces of signature $(n\!-\!1,1)$ at vertex-stabilizer level. To study the Bruhat-Tits strata, we introduce strata models$\!-\!$simpler models that are étale-locally isomorphic to each stratum. They admit two complementary characterizations: (i) as strict transforms under the blow-up of the local model at its worst point, and (ii) via a partial moduli description given by explicit linear-algebraic conditions; from these we deduce smoothness, explicit dimension formulas and irreducibility of the Bruhat-Tits strata.