A nice variety without a smooth model - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:55:21Z http://mathoverflow.net/feeds/question/10463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10463/a-nice-variety-without-a-smooth-model A nice variety without a smooth model Chandan Singh Dalawat 2010-01-02T04:54:14Z 2010-01-10T11:45:06Z <p>Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that</p> <p>--- $X(K)\neq\emptyset$,</p> <p>--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,</p> <p>--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,</p> <p>and yet,</p> <p>--- $X$ is not the generic fibre of any smooth proper $\mathbb{Z}_p$-scheme ?</p> <p><a href="http://mathoverflow.net/questions/416/existence-of-smooth-models/10264#10264" rel="nofollow">My example </a>of a Châtelet surface with these properties is simple enough, but can one do better ?</p> http://mathoverflow.net/questions/10463/a-nice-variety-without-a-smooth-model/11302#11302 Answer by Chandan Singh Dalawat for A nice variety without a smooth model Chandan Singh Dalawat 2010-01-10T08:54:42Z 2010-01-10T11:45:06Z <p>As Minhyong suggests, a curve $C$ (with $C(\mathbb{Q}_p)\neq0$) which has bad reduction but whose jacobian $J$ has good reduction would do the affair. This works because the cohomology of $C$ is essentially the same as that of $J$, and because an abelian $\mathbb{Q}_p$-variety has good reduction if and only if its $l$-adic étale cohomology is unramified for some (and hence for every) prime $l\neq p$ (Néron-Ogg-Shafarevich) or its $p$-adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).</p> <p>I asked Qing Liu for explicit examples. He suggested the curve $$y^2=(x^3+1)(x^3+ap^6)\qquad (a\in\mathbb{Z}_p^\times)$$ when $p\neq2,3$, and $y^2=(x^3+x+1)(x^3+a3^4x+b3^6)$, with $a,b\in\mathbb{Z}_3^\times$, for $p=3$.</p> <p>He refers to Proposition 10.3.44 in his book for computing the stable reduction of these $C$, and to Bosch-Lütkebohmert-Raynaud, <em>Néron models,</em> Chapter 9, for showing that $J$ has good reduction.</p> <p>I "accept" this answer as coming from Minhyong Kim and Qing Liu.</p> http://mathoverflow.net/questions/10463/a-nice-variety-without-a-smooth-model/11304#11304 Answer by norondion for A nice variety without a smooth model norondion 2010-01-10T09:04:32Z 2010-01-10T09:04:32Z <p>Chandan, can you tell me the numbers of the theorems in Bosch-Lütkebohmert-Raynaud?</p>