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Given a regular ring $A$ with an element $t$, consider the "generalised" semistable model
$S := A[X_1,...,X_n]/(P_1\cdots P_n - t)$ over $A$, where $P_i := {X_i}^{e_i}$ and $e_i$ are positive integers.

Question: For which $A$, $t$, and $(e_i)$ is $S$ regular?

E.g., when $n=2$ and $A$ is a discrete valuation ring and $t$ is a uniformiser, then it is easy to prove that $S$ is regular when $e_1$ and $e_2$ are both equal to 1. When $n\ge 3$ and $(A,t)$ is a DVR with uniformiser $t$, is there a reference?

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Assuming $\sum e_i > 1$ it is homework for general regular $A$ except if all $e_i$ divisible by common prime $p$ s.t. $A[1/t]$ has prime of residue char. $p > 0$ with $A$ of char. 0 (hint: regularity of $A/(t)$ is relevant). Exceptional case equivalent to instances of $R[x]/(x^{p^c} - u)$ for $c \ge 1$ and regular local $R$ of mixed char. $(0,p)$ and unit $u$ s.t. $x^p - u$ is irred. over frac. field of $R$ and $u$ a $p$-power in residue field. Then no clean answer; try 1-units in dvrs (write $x = 1+y$). Please work out details of above for yourself; it is far more instructive that way. –  BCnrd Apr 7 '10 at 11:35
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1 Answer

Thank you for your reply!

BCnrd: Do you mean the sum of the $e_i$ is $>1$, or rather $e:=gcd(e_i)$ invertible in $A$?

Granted, some aspects of the problem are homework, for instance when $A$ and $A/(t)$ are regular and $e$ is invertible, then it's standard, Jac. criterion and the standard prop.'s of regular rings. What's really being asked is how does Reg(A) compare to Reg(S) for a given vector of positive integers (e_i) and $t \in A$. Even if $e$ is not invertible in $A$, the regularity of the locus of V(t) in $S$ is easy. So then the interesting locus is $V(e)$ when it is not empty.

So assume the prime in $Spec(S)$ belongs to $D(t)$ and $V(e)$. Restrict to the case initially where $A$ is regular and $t$ is a unit. How does one get to those exceptional cases?

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@Lutz, I think we say the same thing: focus on regular $A$ and $t$ a unit, only problem at pts upstairs over which $A$ has residue char. $p > 0$ dividing $e$, say $e_i = p^c e'_i$ with $c > 0$ and some $e'_i$ not divisible by $p$. Then $S = (A[x]/(x^{p^c} - t))[Y_1,\dots,Y_n]/(\prod Y_i^{e'_i} - x)$; note $x$ is unit in $A[x]/(x^{p^c}-t)$. Ring $S$ is f. flat over $A[x]/(x^{p^c}-t)$, so if $S$ is regular then $A[x]/(x^{p^c}-t)$ must be regular (!!). That brings us to study the latter ring, whose regularity or not is easy away from exceptional case (if I'm not mistaken). Does that clarify it? –  BCnrd Jun 18 '10 at 16:04
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