# Is Krämer's local model for ramified unitary groups isomorphic to the blow-up of Pappas' flat model at the singular point?

I am reading the following two papers:

• Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, 1999 (it seems to be difficult to find online nowadays - only a .ps file remains available),
• Krämer, Local models for ramified unitary groups, 2003.

Let $$n\geq 1$$ and consider integers $$r,s\geq 0$$ such that $$r+s = n$$. Let $$K_0$$ be a $$p$$-adic field with $$p\not = 2$$, and let $$K$$ be a ramified quadratic extension of $$K_0$$. The (naive) local model of signature $$(r,s)$$ associated to a $$K/K_0$$-hermitian space of dimension $$n$$ is a projective scheme $$M_{r,s}$$ over $$\mathcal O_K$$, defined by an explicit moduli problem of linear algebra. In general, eg. when $$|r-s|> 1$$, it is not flat. To rectify this, Pappas introduces the local model $$M_{r,s}^{\mathrm{loc}}$$ by a slight modification of the moduli problem. It comes along with a closed immersion $$M_{r,s}^{\mathrm{loc}}\hookrightarrow M_{r,s}$$. In the special case $$(r,s) = (n-1,1)$$, Pappas proves that $$M_{n-1,1}^{\mathrm{loc}}$$ is normal, Cohen-Macaulay, flat over $$\mathrm{Spec}(\mathcal O_K)$$ and smooth outside of a single closed point $$y\in M_{n-1,1}^{\mathrm{loc}}$$ of the special fiber.

In both papers, the authors consider the setting as above. They then build a resolution of the singularity of $$M_{n-1,1}^{\mathrm{loc}}$$, but a priori in two different ways:

• Pappas considers the blow-up $$\mathrm{Bl}(M_{n-1,1}^{\mathrm{loc}})$$ of $$M_{n-1,1}^{\mathrm{loc}}$$ at $$y$$.
• Krämer builds a new local model $$\mathcal M_{n-1,1}$$ by adding extra datum to the moduli problem defining $$M_{n-1,1}^{\mathrm{loc}}$$.

Both constructions, $$\mathrm{Bl}(M_{n-1,1}^{\mathrm{loc}})$$ and $$\mathcal M_{n-1,1}$$, give rise to projective regular schemes over $$\mathcal O_K$$ with semistable reduction. They both come with a natural map to the local model $$M_{n-1,1}^{\mathrm{loc}}$$, inducing an isomorphism outside of the fiber over $$y$$.

It seems very likely that both constructions are actually isomorphic, ie. $$\mathcal M_{n-1,1} \simeq \mathrm{Bl}(M_{n-1,1}^{\mathrm{loc}})$$. However, this does not seem to be stated anywhere in the literature, as far as I can tell. Besides, Krämer refers to Pappas' work by saying that

First we have to resolve the singularities of the local model. In [Pappas], this was done by blowing up the singular locus. Our approach is different: We define a resolution $$\mathcal M_{n-1,1} \to M_{n-1,1}^{\mathrm{loc}}$$ by posing a moduli problem analogous to the Demazure-resolution of a Schubert variety in the Grassmannian.

In other words, Krämer seems to emphasize that this construction a priori differs from Pappas' one. If they really did give isomorphic resolutions, one would expect a mention of this fact at least in the introduction...

Thus, I'd like to ask anyone who would be familiar with local models. Are the two above constructions known to be isomorphic or not?

Proposition 2.2. $$N^{\text{Kra}}$$ is the blow-up of $$N^{\text{Pap}}$$ along its singular locus Sing.