7
$\begingroup$

The affine scheme $U := \mathrm{Spec}(\mathbb{Z}_p[x_0, \ldots, x_d]/(x_0\cdots x_d - p))$ is regular and is a basic example of a semistable scheme over $\mathbb{Z}_p$. How does one build a proper regular $\mathbb{Z}_p$-scheme $X$ that contains $U$ as an affine open and is semistable over $\mathbb{Z}_p$ (in the sense that its special fiber is a normal crossings divisor in $X$)?

Of course, according to various resolution of singularities type conjectures, such $X$ should exist for vastly more general $U$. How does one build it in this particular simple case? Simply homogenizing the equation does not lead to a regular $X$ if $d > 1$.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ I have this vague idea that this is the kind of question that the theory of weighted projective spaces can sometimes deal with in a very slick manner. But weighted projective spaces can have mild singularities and so one would have to understand the details better than I do to know whether this comment is going anywhere. The other thing that instantly sprang to mind is: if you invert all but one of the $x_i$ then you can throw away the last one $x_j$ and you get an open in affine space with coordinates $x_i^{-1}$, $i\not=j$. Can you glue these affine spaces together? Details need checking :-/ $\endgroup$
    – Kevin Buzzard
    Commented Feb 8, 2017 at 8:54
  • 1
    $\begingroup$ I suspect the following works. Begin with $(\mathbb{P}^1_{\mathbb{Z}_p})^d = \mathbb{P}^1_{\mathbb{Z}_p}\times_{\text{Spec}\ \mathbb{Z}_p}\dots \times_{\text{Spec}\ \mathbb{Z}_p}\mathbb{P}^1_{\mathbb{Z}_p}$ with homogeneous coordinates $([s_0,t_0],[s_1,t_1],\dots,[s_d,t_d])$. Let $X'$ be the hypersurface with defining equation $s_0\cdots s_d - p t_0\cdots t_d$. Now iteratively blowup the following loci. First blowup the ideal $\langle y_i y_1 \rangle_{0\leq i < j \leq d}$, then blowup the strict transform of the ideal $\langle y_iy_jy_k \rangle_{0\leq i<j<k\leq d}$, etc. $\endgroup$
    – Jason Starr
    Commented Feb 8, 2017 at 13:21
  • 1
    $\begingroup$ Since the blowing up in $(\mathbb{P}^1)^d$ of that sequence of ideals (beginning with the ideal $\langle y_i \rangle_{0\leq i\leq d}$ that is disjoint from our hypersurface) is a smooth toric variety, it should be straightforward to compute the strict transforms of $s_0\cdots s_d - p t_0\cdots t_d$ on the basic open affine spaces in this toric variety. That should settle whether or not this blowing up is regular. $\endgroup$
    – Jason Starr
    Commented Feb 8, 2017 at 13:31
  • $\begingroup$ Thank you for your comments. I think, I found an answer by looking at the proof of Prop. 2.1 of math.uni-muenster.de/u/urs.hartl/Publikat/SemistBC.pdf ; namely, one first does the case $d = 1$ by homogenizing the equation to $x_0x_1 - py^2$ to get a projective semistable $X_1$. Then one takes the $d$-fold fiber product $X_1^d$ over $\mathbb{Z}_p$, follows the linked proof to desingularize it, and observes from that proof that $U$ does indeed occur in the desingularization. $\endgroup$
    – Lisa S.
    Commented Feb 8, 2017 at 16:58

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .