On the moduli description of ramified unitary local models of signature $$(n-1,1)$$

Abstract We provide a moduli description of the ramified unitary local model of signature $$(n-1,1)$$ with arbitrary parahoric level structure, assuming the residue field has characteristic not equal to 2, thereby confirming a conjecture of Smithling (Int Math Res Not (24) 2015:13493–13532, 2015). Our approach involves writing down explicit equations for the special fiber and proving that they define a normal, Cohen–Macaulay scheme, which is also of independent interest. As applications, we obtain moduli descriptions for: (1) ramified unitary Pappas–Zhu local models with arbitrary parahoric level; (2) the irreducible components of their special fiber in the maximal parahoric case; (3) integral models of ramified unitary Shimura varieties with arbitrary (quasi-)parahoric level.