text:
\section{Introduction}

The main goal of this work is to establish a version of semistable reduction (or relative desingularization) of schemes over a base of the form $B_n=\Spec(k[\pi]/(\pi^n))$, where $k$ is a field of characteristic zero. In particular, we were asked about existence of such a result by Christian Schnell and Junchao Shentu, and an answer to this problem might have applications to Hodge theory. A general theory of resolution of relative (log) morphisms $X\to B$ was recently developed in \cite{ATW} under the assumption that $B$ is log regular, so it cannot be directly applied to our case. There is a way to generalize that theory to the base we need, but it will be worked out elsewhere, and in this work we decided to use more standard tools. In fact, we manage to deduce our main result from the classical resolution and principalization theorems.

\subsection{Log smoothness over thick points}
We would like to stress that a priori it is not even so clear how to formulate an appropriate semistable reduction conjecture whose goal is to resolve a generically smooth morphism $X\to B=B_n$. For comparison, recall the situation with semistable reduction over a trait, say over $S=\Spec(R)$ with $R=k[\pi]_{(\pi)}$. If $Z\to S$ has a connected but not irreducible closed fiber $Z_s$, for example, $Z=\Spec(R[x,y]/(xy-\pi))$, then by Zariski's connectedness theorem the same will be true for any modification $Z'$ of $Z$, and hence $Z'\to S$ is not smooth. The best one can do is to resolve $Z$ and make $Z_s$ an snc divisor. In the zero characteristic case this improves the morphism $Z\to S$ drastically: \'etale-locally $Z$ becomes isomorphic to the model charts of the form $\Spec(R[t_1\.t_n]/(t_1^{d_1}\dots t_n^{d_n}-\pi))$, in particular, it is log smooth with respect to the natural closed fiber log structures. For the sake of completeness, we note that such a morphism is called semistable if $d_i\in\{0,1\}$, and one can further make $Z\to S$ semistable by an extension of $R$ and log (or toroidal) blowings up of $Z$, but we will not need this.

Adopting the logarithmic approach is also absolutely critical for the results of \cite{ATW}, so it is natural to adopt it in our case too and to provide $B$ with the log structure induced by $\bbN\log\pi$. Nevertheless, it turns out that one cannot in general modify a generically smooth $B$-scheme $X$ to $X'$ which is log smooth over $B$. Here is the simplest example: $X=\Spec(k[x,y]/(y^n))$ with the morphism $X\to B$ given by $\pi=xy$. Indeed, the morphism is not smooth at $x=0$ because the nilradical of $\calO_X$ is not generated by $\pi$ at this point. The situation does not improve if we increase the log structure on $X$ by adding $\bbN\log x$. Moreover, any modification $X'\to X$ induces an isomorphism of the reductions and only increases the nilpotent structure (for example, by replacing $y$ by $y'=y/x^n$), hence it does not improve the situation either. The only way to achieve some sort of resolution of $X\to B$ is to use the whole class of smooth $B$-curves, including the reducible ones, and there are plenty of them. For example, $Y=\Spec(k[x,y]/(x^ny^n))$ is log smooth over $B$ (where $\pi=xy$), and $X$ is an irreducible component of $Y$. In fact, this example is the closed fiber of the morphism $Z\to S$ from the example above, but reducible closed fibers over $S$ show up naturally, while reducible schemes over $B$ at first glance look as a pathology, and by no means they can show up as modifications of irreducible ones.

We hope that the above discussion and relation to the classical semistable reduction over $S$ give enough motivation for our definition: a desingularization of a $B$-scheme $X$ is a modification $X'\to X$ such that locally $X'$ can be realized as an irreducible component and a monomial subscheme of a log smooth $B$-scheme $Y$. The main result of the paper -- Theorem~\ref{mainth1} proves that any generically smooth $B$-scheme $X$ possesses such a desingularization. Moreover, we show that it can be chosen of a special combinatorial type: $X'$ is \'etale over $\Spec(R[\veps,t_1\.t_n]/(\veps^n))$ with $\pi$ mapped to $\veps t_1^{d_1}\.t_n^{d_n}$ and the log structure generated by $\veps$ and $t_1\.t_n$. In addition, we show that any closed subscheme $Z\into X$ can be made a monomial subscheme of $X'$, which is an analogue of resolution of pairs.

\begin{rem}
Here is another argument in favor of our definition of desingularization over $B$. We mentioned that principalization of ideals on log smooth $B$-schemes is possible (and will be developed elsewhere). As usually, it can be used to obtain a desingularization of $B$-schemes $X$ as follows. Separate components, reducing to the case when $X$ is irreducible. Locally embed $X$ into a log smooth $B$-scheme $M$. Principalize $\calI_X$ on $M$ by blowing up centers which are monomial or log smooth over $B$ and stop when the new blowing up center $V$ contains the generic point of the strict transform $X'$ of $X$. In particular, $V$ is log smooth over $B$ and $X'$ is its irreducible component. In the situation of \cite{ATW} (similarly to the classical resolution) the connected components of $V$ are irreducible, hence $X'$ is one of them and $X'\to X$ is a $B$-resolution. In our situation, a log smooth $V$ does not have to be irreducible, hence we only obtain a modification $X'\to X$ which is an irreducible component of a log smooth $V$. This is precisely the above definition of a $B$-desingularization.
\end{rem}

\begin{rem}
(i) One can wonder if there exist another type of resolutions: a proper morphism $X'\to X$ which is birational on one component of $X'$ and such that $X'\to B$ is log smooth. We do not study this question in the paper, but it seems plausible that in general this is impossible.

(ii) One can also ask if the embedding of $X'$ as an irreducible component of a $B$-log smooth $Y$ can be made globally. This also seems to be a not so natural question, though we show in the paper that this is possible, and if $X\to B$ is proper, then $Y$ can also be taken to be proper over $B$. This construction uses log blowings up, which are natural non-birational proper morphisms $X'\to X$ (which contract components). However, log blowings up cannot be used to attack the question in (i) because they are log \'etale, and hence if $X\to B$ is not log smooth, then the same is true for $X'\to B$.
\end{rem}

\subsection{Outline of the paper}
Now, let us briefly describe our other results and the structure of the paper. In Section~\ref{thicksec} we extend some resolution results to non-reduced schemes. For the sake of completeness we recall in \S\ref{Hirsec} Hironaka's resolution of non-reduced schemes by {\em thick manifolds} -- schemes $X$ with a regular reduction $\tilX=X^\red$ and normally flat along $\tilX$. However, we only need to work with schemes $X$ whose nilradical is generically principal, and their resolution is deduced in \S\ref{desingsec} from the classical functorial principalization: in Theorem~\ref{nonembeddedth} we show that any such $X$ possesses a functorial modification $X'$, which is a principally thick manifold or a ptm -- a thick manifold with a locally principal nilradical, see \S\ref{ptmsec}.

In \S\ref{princsec} we deduce principalization of subschemes of a ptm $X$ from the classical principalization of subschemes of $\tilX$, and in \S\ref{Bschemes} we prove our main result on resolution of varieties over $B$. As in the case of $S$-schemes, we just resolve $X$ (by a thick manifold since $X$ is non-reduced) and then principalize the divisor $(\pi)$ on $X$. There are small additional subtleties in this case, which are dealt with in \S\ref{monomsec}. Finally, in \S\ref{othersec} we explain how analogous results can be proved for certain quasi-excellent schemes, formal schemes and analytic spaces.

Section \ref{mainsec} is devoted to proving that in the case of varieties, after an additional work the resolved $B$-scheme $X'$ can be globally embedded into a log smooth $B$-scheme $Y$. This operation is less functorial, it depends on the choice of a retract, and our argument does not apply to general qe schemes or analytic varieties, because retract does not have to exist in these cases. It might be the case that the generalization of our main result we prove in Theorem~\ref{mainth} addresses a not so natural question. Still, the intermediate results we prove about retracts and factorization of birational morphisms of non-reduced schemes can be of an independent interest.

\section{Desingularization on principally thick manifolds}\label{thicksec}
In this section we will extend the classical resolution theory to schemes which do not have to be generically reduced but whose nilradical is generically principal. Everything will be deduced rather straightforwardly from the classical theory.

\subsection{Hironaka's resolution of non-reduced schemes}\label{Hirsec}
In the famous 1964 paper \cite{Hironaka} Hironaka also established desingularization of arbitrary non-reduced varieties that we are going to recall briefly.

\subsubsection{Thick manifolds}
Let us fix some terminology first. We warn the reader that the notion of thick manifolds is non-standard.

\begin{defin}\label{thickdef}
(i) A scheme $X$ is {\em normally flat} along a subscheme $Y$ given by an ideal $\calI\subseteq\calO_X$ if $\calI^n/\calI^{n+1}$ is a flat module over $\calO_Y=\calO_X/\calI$ for any $n$.

(ii) A locally noetherian scheme $X$ is called a {\em thick manifold} if its reduction $X^\red$ is regular and $X$ is normally flat along $X^\red$.

(iii) By {\em thickness} of a noetherian scheme $X$ at a point $x$ we mean the sum $h_X(x)$ of dimensions of the $k(x)$-vector spaces $(\calN^n/\calN^{n+1})\otimes k(x)$, where $\calN$ is the nilradical of $\calO_X$.
\end{defin}

The normal-flatness condition means that although there might be a non-reduced structure, it behaves ``as nice as possible''.

\begin{lem}\label{thicklem}
(i) If $\eta\in X$ is a generic point, then $h_X(\eta)$ is the length of the Artin ring $\calO_{X\eta}$.

(ii) The thickness function $h_X\:X\to\bbN$ is upper-semicontinuous, and $X$ is normally flat along $X^\red$ if and only if $h_X$ is a locally constant function. In the latter case, $X$ has no embedded components.
\end{lem}
\begin{proof}
The first claim is obvious. In (ii) set $Y=X^\red$ and note that for a finite $\calO_Y$-module $M$ the function $r_M(x)=\dim_{k(x)} M\otimes k(x)$ is upper semicontinuous on $Y$, and $r_M$ is locally constant if and only if $M$ is flat. The equivalence in (ii) follows by applying this to $h_X=\sum_nr_{\calN^n/\calN^{n+1}}$ -- due to the semicontinuity, $h_X$ is locally constant if and only if each summand is. Finally, if $X$ has an embedded component, then there exists a closed immersion $X'\into X$ such that the locus $V$ where $X'\neq X$ is nowhere dense but non-empty. In this case, $h_X=h_{X'}$ outside of $V$ and $h_{X}$ jumps on $V$. In particular, for any generic point $\eta$ with a specialization $x\in V$ one has $h_X(\eta)=h_{X'}(\eta)\le h_{X'}(x)<h_X(x)$ and hence $X$ is not a thick manifold.
\end{proof}

\subsubsection{Hironaka's theorem}\label{Hironaka}
In a sense, thick manifolds are non-reduced schemes which are as close to regular ones as possible, and desingularization of non-reduced schemes by thick manifolds is the best one can hope for -- one has to keep the nilpotent structure at the generic points, but it becomes as simple as possible everywhere else.

\begin{defin}\label{moddef}
(i) Throughout this paper a morphism $X'\to X$ is called a {\em modification} if it is proper and induces an isomorphism $U'=U$ of dense open subschemes.

(ii) A {\em desingularization} of a scheme $X$ is a modification $X'\to X$ such that $X'$ is a thick manifold.
\end{defin}

Hironaka proved that this desingularization is, indeed, possible.

\begin{theor}[{\cite[Main Theorem $\rm I^*$]{Hironaka}}]
There exists a construction which associates to any variety $X$ of characteristic zero a projective desingularization $\calF(X)=X'\to X$
\end{theor}

In fact, Hironaka proved his theorem for the more general class of schemes of finite type over a local quasi-excellent ring of characteristic zero.

\subsubsection{Strong resolution}
Hironaka's proof is existential, but modern methods that grew from his approach can also provide a smooth-functorial desingularization. Unfortunately, a convenient reference is missing in the literature, so let us describe the situation briefly. Nowadays there are two main resolution methods, both originating from Hironaka's work, that will be called {\em basic} (or weak in the literature) and {\em strong}. Basic resolution is achieved by blowing up smooth centers in the ambient manifold $M$, but the induced blowings up of the scheme $X$ itself may have singular centers. Its primary invariant is the order of an ideal $I_X\subset\calO_M$. This resolution was studied in the majority of recent works and it is smooth-functorial by results of W{\l}odarczyk, see \cite{Wlodarczyk}.

Strong resolution uses the whole Hilbert-Samuel function as the primary invariant and it only blows up smooth centers such that $X$ is normally flat along them. The price one has to pay is that the process is much heavier, as well as the proofs. In fact, one reduces this resolution to the basic one by encoding the HS function in terms of the order of another ideal (called a presentation). So far, the only known way to resolve general non-reduced schemes is via a strong resolution methods. A canonical strong desingularization was established by Bierstone-Milman, see \cite[Theorem~11.14]{Bierstone-Milman} (though it was only formulated in the embedded version), and its functoriality properties were briefly checked in \cite[Theorem 6.1. Addendum]{BMT} (one should check that the reduction to a basic method is smooth-functorial).

However, we will see that already the basic method suffices to resolve varieties with generically principal nilradical, and this is the only case we will need later. In particular, smooth-functoriality is then automatic.

\subsection{Principally thick manifolds}\label{ptmsec}
We say that a scheme $X$ is {\em nilprincipal} if its nilradical is a locally principal ideal. If, in addition, $X$ is a thick manifold, then we say that $X$ is a {\em principally thick manifold} or a {\em ptm}.

\begin{defin}
Let $X$ be a ptm. By a {\em family of parameters} at a point $x\in X$ we mean a family $\veps,t_1\.t_d\in\calO_{X,x}$ such that $\veps$ generates the nilradical of $\calO_{X,x}$ and $t_1\.t_d$ induce a family of regular parameters of $\calO_{X^\red,x}=\calO_{X,x}/(\veps)$. We call $\veps$ a {\em nilpotent parameter} and call $t_1\.t_d$ {\em regular parameters}.
\end{defin}

Thickness on a ptm has especially simple interpretation.

\begin{lem}\label{thickness}
Let $X$ be a ptm, $x\in X$ a point and $\veps\in\calO_x$ a nilpotent parameter at $x$. Then the thickness $h_X(x)$ is the minimal number $h$ such that $\veps^h=0$.
\end{lem}
\begin{proof}
Let $\calN$ be the nilradical, then $\calN_x=\veps\calO_x$ and $\calN^n/\calN^{n+1}$ is non-zero if and only if $\calN^n\neq 0$, that is $n<h$. In the latter case, $(\calN^n/\calN^{n+1})\otimes k(x)$ is generated by the image of $\veps^n$ and hence is of dimension 1.
\end{proof}

\begin{rem}
(i) Due to our definition, ptm's of thickness 1 are just usual manifolds (or regular schemes) with dummy nilpotent parameter $\veps=0$. Because of this they should often be dealt with slightly differently, and since all results for them are standard we will simply exclude this case when this is convenient.

(ii) We will later enhance ptm's to log schemes with the log structure generated by $\log(\veps)$. In this situation, the case of thickness 1 is not exceptional anymore, in particular, the corresponding log scheme is not log regular.
\end{rem}

\begin{lem}\label{cotangentlem}
If $X$ is a non-reduced ptm and $\veps,t_1\.t_d$ is a family of parameters at $x$, then their images form a basis of the cotangent space $m_x/m_x^2$.
\end{lem}
\begin{proof}
Since $X$ is singular and of dimension $d$ at $x$, we have $\dim(m_x/m_x^2)>d$. The $d+1$ parameters generate $m_x$, hence their images generate $m_x/m_x^2$ and by a dimension consideration they form a basis.
\end{proof}

\begin{lem}\label{thickmanlem}
Let $M$ be a regular scheme and $X\into M$ a closed subscheme given by an ideal $\calI\subset\calO_M$, $x\in X$, $\dim_x(M)=d$ and $\dim_x(X)=l$. Then $X$ is a non-reduced ptm locally at a point $x\in X$ if and only if there exist a family of local parameters $x_1\.x_d\in\calO_{M,x}$ such that $\calI_x=(x_1^n,x_2\.x_r)$ for $r=d-l$. In addition, if these conditions are satisfied and $(\veps,t_1\.t_l)$ is a family of parameters of $\calO_{X,x}$, then one can choose $x_1,x_{r+1}\.x_d$ to be any lift of $(\veps,t_1\.t_l)$.
\end{lem}
\begin{proof}
The inverse implication is obvious since $X^\red$ is locally given by the vanishing of $(x_1\.x_r)$, and the nilradical of $\calO_{X,x}$ is generated by the image of $x_1$.

Conversely, assume that $X$ is a ptm and fix any family $(x_1,x_{r+1}\.x_d)$ of elements of $\calO_{M,x}$ that reduces to a family of parameters $(\veps,t_1\.t_l)$ at $x$. The images of $(x_1,x_{r+1}\.x_d)$ in $m_{M,x}/m_{M,x}^2$ are linearly independent because their images in $m_{X,x}/m_{X,x}^2$ form a basis by Lemma~\ref{cotangentlem}. The kernel of the map $m_{M,x}/m_{M,x}^2\onto m_{X,x}/m_{X,x}^2$ is the image of $\calI_x$, hence the family $(x_1,x_{r+1}\.x_d)$ can be completed to a family of regular parameters $(x_1\.x_d)$ so that $x_2\.x_r\in\calI_x$.

Let $n$ be the thickness of $X$ at $x$. Then locally at $x$ the ideal $\calI'=(x_1^n,x_2\.x_r)$ is contained in $\calI$, hence defines a ptm $X'$ containing $X$. We claim that the surjection $\phi\:\calO_{X',x}\onto\calO_{X,x}$ is an isomorphism. Indeed, any element $a\in\Ker(\phi)$ is nilpotent because $X^\red$ and $X'^\red$ are smooth of the same dimension $l=d-r$ at $x$, and using that $X$ and $X'$ are of thickness $n$ we obtain that $a=0$.
\end{proof}

We say that a ptm (or a scheme) $X$ is {\em locally embeddable (in a regular scheme)} if any point $x\in X$ has a neighborhood isomorphic to a closed subscheme of a regular scheme $M$. For example, any $k$-variety satisfies this property. The above lemma implies that locally embeddable ptms are isomorphic to a divisor $nD$ with a regular $D$ in an ambient regular scheme:

\begin{cor}\label{thickmancor}
If $X$ is a locally embeddable ptm, then any point $x\in X$ possesses a neighborhood $U$ and a closed immersion $U\into M$, such that $M$ is regular and $U$ is the divisor in $M$ of the form $V(t^n)$, where $t$ is a parameter of $\calO_{M,x}$ and $n$ is the thickness of $X$ at $x$.
\end{cor}
\begin{proof}
By Lemma~\ref{thickmanlem}, $x$ has a neighborhood $U$ isomorphic to a closed subscheme $V(t_1^n,t_2\.t_r)$ of a regular scheme $M'$, where $t_1\.t_d$ is a regular family of parameters of $\calO_{M',x}$. It remains to take $M=V(t_2\.t_r)$ and define $t\in\calO_{M,x}$ to be the image of $t_1$.
\end{proof}

The following corollary certainly holds for arbitrary ptm's, but we only need the case of locally embeddable ones and then the argument is very simple.

\begin{cor}\label{CMcor}
Any ptm $X$, which is locally embeddable in a regular scheme, is Cohen-Macaulay. In particular, if $i\:U\into X$ is an open immersion, whose complement is of codimension at least 2, then $\calO_X=i_*\calO_U$.
\end{cor}
\begin{proof}
By Corollary \ref{thickmancor}, locally $X$ is a Cartier divisor in a smooth (hence CM) variety. Therefore, $X$ is itself CM. The second claim is, perhaps, better known for normal schemes, but it holds more generally for $S_2$-schemes and, in fact, provides an equivalent characterization of this property. The original source is the theory of $Z$-purity, see \cite[Th\'eor\`eme~IV.5.10.5]{ega}, a modern treatment (in the relative setting) is in \cite[Proposition~3.5]{Hassett-Kovacs}.
\end{proof}

\subsubsection{Other characterizations}
For the sake of completeness we provide alternative characterizations of ptm's, though this will not be used in the sequel.

\begin{lem}\label{nilprincipallem}
Given a scheme $X$ the following conditions are equivalent:

(i) $X$ is a ptm.

(ii) $X$ is a generically nilprincipal thick manifold.

(iii) $X^\red$ is regular, $X$ is nilprincipal and without embedded components.
\end{lem}
\begin{proof}
(i) implies (ii) and (iii) because thick manifolds have no embedded components by Lemma~\ref{thicklem}(ii). In the sequel, let $\calI$ denote the nilradical of $\calO_X$.

(ii)$\Longrightarrow$(i) Looking at the generic points we see that the locally free module $\calI/\calI^2$ is of rank 1. Its local generator at a point $x$ lifts to a local generator of the stalk $\calI_x$ and hence $X$ is nilprincipal.

(iii)$\Longrightarrow$(i) We will work locally at $x\in X$. Let $\veps\in\calO_x$ be a generator of $\calI_x$ and let $n$ be the minimal number with $\veps^n=0$. We should prove that $X$ is normally flat along $X^\red$ at $x$ and we claim that, in fact, each $\calI_x^i/\calI_x^{i+1}$ with $i<n$ is free of rank 1 over $\calO_{X^\red,x}=\calO_{X,x}/\calI_x$. Clearly, $\calI_x^i/\calI_x^{i+1}$ is generated by the image of $\veps^i$, so we should prove that its annihilator in $\calO_{X,x}/\calI_x$ is zero. Moreover, it suffices to prove the latter only for $i=n-1$, and since $\calI_x^n=0$ this happens if and only if the annihilator of $\veps^{n-1}$ in $\calO_{X,x}$ is contained in (in fact, equal to) $\calI_x$. The latter holds because otherwise $X$ would contain an embedded component on whose complement $\veps^{n-1}$ vanishes.
\end{proof}

\begin{rem}
Note that $X$ is generically nilprincipal if and only if for any generic point $\eta\in X$ the Artin ring $\calO_\eta$ has principal maximal ideal $m_\eta=(\veps)$. Such Artin rings are very special, e.g. see \cite[Proposition~8.8]{AM}, in addition, they can be described as quotients of DVR's by non-zero ideals.
\end{rem}

\subsection{Desingularization of generically nilprincipal varieties}\label{desingsec}
Now we are ready to formulate the resolution result for non-reduced schemes with simplest nilpotent structure.

\begin{theor}\label{nonembeddedth}
There exists a construction which associates to any generically nilprincipal variety $X$ of characteristic zero a projective desingularization $\calF(X)=X'\to X$ which depends on $X$ smooth-functorially: for any smooth morphism $Y\to X$ one has that $\calF(Y)=\calF(X)\times_XY$.
\end{theor}
\begin{proof}
Step 1. {\em Assume that $X$ is irreducible and embedded as a closed subscheme in a smooth variety $M$.} In this case we apply the smooth-functorial principalization from \cite{Wlodarczyk} to $\calI_X\subset\calO_M$ (one can use also the methods of \cite{Kollar} or \cite{Bierstone-Milman-funct}, but they all are equivalent). It outputs a sequence of blowings up $f_i\:M_i\to M_{i-1}$, $1\le i\le n$ with smooth centers $V_i\subset M_i$ and snc boundaries $E_i\subset M_i$ such that $E_0=\emptyset$, $M_0=M$, $E_i=f_i^{-1}(E_{i-1})\cup f_i^{-1}(V_{i-1})$, $V_i$ has simple normal crossings with $E_i$ and lies in the preimage of $X$, and $\calI_X\calO_{M_n}$ is invertible and supported on $E_n$. As one always does when deducing resolution from principalization, we cut this sequence at $M_l$ such that $f_{l+1}$ is the first blowing up whose center $V_l$ maps onto $X$; it exists because the preimage of $X$ lies in $E_n$. Then the strict transform $X_l\subset M_l$ of $X$ (i.e. the schematic closure of $X\times_MM_l\setminus E_l$) is non-empty, lies in $V_l$ and contains an irreducible component of $V_l$. Since $V_l$ is smooth, $X^\red_l$ is an irreducible component of $V_l$, in particular, it is smooth. So far, we repeated the standard argument, which constructs the resolution $X_l^\red\to X^\red$ of the integral scheme $X^\red$, but we claim that, moreover, $X_l\to X$ is a resolution of $X$.

Generically on $X$ the principalization works as follows: there is no boundary and whenever $\dim(M)>\dim(X)+1$ the order is one by Lemma~\ref{thickmanlem}, so one simply replaces $M$ by a maximal contact -- any smooth divisor containing $X$. Once $\dim(M)=\dim(X)+1$, one has $\calI_X=(t^n)$, where $n$ is the thickness of $X$ and $H=V(t)$ is the reduction of $X$. It follows that the order is $n$, the only maximal contact is $H$ and the algorithm simply resolves the marked ideal $(\calI_X,n)$ by blowing up $H$. By our assumption this happens at stage $l$, so at this stage at any point $x\in X_l$ there is no new boundary (we are in Step 1b of the algorithm in \cite{Wlodarczyk}), $d=\ord_x(\calI_{X_l})$ equals 1 whenever $\dim(M_l)>\dim({X_l})+1$, and $d=n$ once $\dim(M_l)=\dim({X_l})$ because the algorithm first deals with the locus where $\dim({X_l})$ is maximal and by our assumption it blows up the whole $H={X_l}^\red$. Note that $nH$ is the schematic closure of the generic point of ${X_l}$, hence $nH\into {X_l}$ and $\calI_{X_l}$ is contained in the divisorial ideal $\calI_H^n$. In particular, $\calI_H^{-n}\calI_{X_l}$ is an ideal, and since the degrees of $\calI_H^n$ and $\calI_{X_l}$ are equal to $n$ everywhere along ${X_l}$, the order of $\calI_H^{-n}\calI_{X_l}$ is zero, that is, $X_l=nH$. So, as claimed $X_l$ is a principally thick manifold and $X_l\to X$ is a projective desingularization.

For the sake of completeness, we note that the blowing up $X_{l+1}\to X_l$ is the last blowing up of the principalization, it happens at Step  1ba in \cite{Wlodarczyk}, and the invariant at this step is $(1,0,1,0\.1,0,n,0,\infty)$.

Step 2. {\it Assume that $X$ is irreducible.} In this case one locally embeds $X$ into a smooth variety $M$ and takes the resolution $X_l\to X$ provided by Step 1. A standard argument used in all works on functorial resolution shows that this is independent of the local choice of embeddings and globalizes to a smooth-functorial method to resolve $X$ -- usually one works with an integral $X$ because principalization does not provide resolution of general non-reduced varieties, but this reduction only uses that $X$ is irreducible.

Step 3. {\it The general case.} In general, let $\eta_1\.\eta_s$ be the generic points of $X$ with the induced scheme structure and let $X_i$ be the schematic closure of $\eta_i$. Then $\coprod_iX_i\to X$ is a modification, which depends on $X$ in a way which is easily seen to be compatible with smooth morphisms $X'\to X$. If $Y_i\to X_i$ are the resolutions provided by Step 2, then $\calF(X)=\coprod_iY_i$ is a resolution of $X$, and by the above $\calF$ is smooth-functorial.
\end{proof}

\section{Resolution on ptm's}\label{onptm}
In this section we will prove our main result -- a version of semistable reduction over closed subschemes of a trait (i.e. zero dimensional schemes with principal radical). In the sequel, given a ptm $X$ we will use the notation $\tilX=X^\red$ to denote its reduction.

\subsection{Principalization on ptm's}\label{princsec}
In this section we construct principalization of ideals on a ptm using the usual principalization on its reduction.

\subsubsection{Blowings up}
Principalization is achieved by iteratively blowing up regular centers.

\begin{lem}\label{blowuplem}
Assume that $X$ is a ptm and $V\into X$ a regular center. Then $X'=\Bl_V(X)$ is a ptm too, and the reduction of $X'$ is the blowing up of the reduction of $X$ along $V$, that is, $\tilX'=\Bl_V(\tilX)$.
\end{lem}
\begin{proof}
The second claim follows from the fact that $\tilX'$ is the strict transform of the closed subscheme $\tilX$ of $X$, and hence by the classical properties of blowings up $\tilX'$ is the blowing up of $\tilX$ along $V\times_X\tilX=V$. In particular, $\tilX'$ is regular.

It remains to check that $X'$ is a ptm, and this is a local claim over a point $x\in V$. So, we can assume that $X=\Spec(A)$ is a local scheme with closed point $x$. Choose local parameters $\veps,t_1\.t_n$ at $x$, such that $V=V(\veps,t_1\.t_r)$. Then the $\veps$-chart is empty, so $X'$ is covered by the charts $X'_i=\Spec(A_i)$, with $A_i=A[\veps/t_i,t_1/t_i\.t_r/t_i]\subseteq A_{t_i}$, and it is easy to see that $\veps'=\veps/t_i$ generates the nilradical of $A_i$. Thus, the nilradical of $X'$ is locally principal and the reduction $\tilX'$ is regular, that is, $X'$ is a ptm.
\end{proof}

\subsubsection{Snc divisors}\label{boundary}
Working on a ptm $X$ one has to distinguish Weil and Cartier divisors. Weil divisors are just divisors on the reduction, while Cartier divisors contain infinitesimal information as well. Saying a ``divisor'' on $X$ we will always mean an effective Cartier divisor $D=V(f)\subset X$. A divisor $D$ will be called {\em snc} if locally on $X$ it factors as $D=\sum_i D_i$ such that each reduction $\tilD_i=D_i\times_X\tilX$ is smooth and $\tilD=\cup_i\tilD_i$ is an snc divisor on $\tilX$. A divisor $D'$ is $D$-monomial if it is of the form $D'=\sum_i n_iD_i$. A regular closed subscheme $V\into X$ has {\em simple normal crossings with} $D$ if it has simple normal crossings with $\tilD$ as a subscheme of $\tilX$.

\begin{rem}\label{sncrem}
If $D$ is a divisor with an snc reduction $\tilD=\cup_iD_i$, then it can freely happen that $D$ is not snc because the factorization of $\tilD$, does not lift to a factorization of $D$, and the same is true for monomiality. For example, if $X=\Spec(k[x,y,\veps]/(\veps^2))$ then $D=V(xy+\veps)$ is not snc, though $\tilD$ is snc, and $D'=V(y^2+\veps)$ is not $V(y)$-monomial, though $\tilD'$ is $V(y)$-monomial. In addition, there exists different snc divisors with the same reduction, for example, $(x)$ and $(x+\veps)$.
\end{rem}

In the sequel, by a {\em ptm with a boundary} we mean a ptm $X$ and an snc divisor $E\into X$, also called a {\em boundary}. A boundary $E$ is called {\em ordered} if it is provided with a decomposition $E=\sum_{i=1}^lE_i$ into components (not necessarily irreducible) with smooth reductions $\tilE_i$.

\subsubsection{Addmissible blowings up and transforms}
Assume that $(X,E)$ is a ptm with a boundary and $\calI\subseteq\calO_X$ is an ideal corresponding to a closed subscheme $Z=V(\calI)$, then a {\em $Z$-admissible blowing up} is a blowing up $f\:X'=\Bl_V(X)\to X$, where $V$ is regular, has snc with $E$ and is contained in $Z$. Since $\calI\subseteq\calI_V$ and the exceptional divisor $E_f=V\times_XX'$ is a Cartier divisor, the {\em principal transform} $\calI'=(\calI\calO_{X'})\calI_{E_f}^{-1}$ is defined. It corresponds to the subscheme $Z'$ of $X'$ that we also denote $Z\times_XX'-E_f$ and call the {\em principal transform} of $Z$ under $f$.

In addition, if $E=\sum_{i=1}^n E_i$ is ordered, then for each $i$ we set $E'_i=E_i\times_XX'-V_i\times_XX'$, where $V_i$ is the union of all components of $V$ contained in $E_i$ (and so $V_i\times_XX'$ is the component of $E_f$ contained in $E_i\times_XX'$). Finally, we set $E'_{n+1}=E_f$ and define the total transform of $E$ to be the ordered boundary $E'=\sum_{i=1}^{n+1}E'_i$. Since $E'$ is independent of the ordering of $E$ and an ordering can be chosen locally, we also obtain a definition of total transform for unordered boundaries.

When the thickness is 1 these definitions are nothing but the classical ones from the principalization theory on regular schemes. Moreover, they are compatible with the reduction:

\begin{lem}\label{transformlem}
Assume that $(X,E)$ is a ptm with an (ordered) boundary, $Z\into X$ is a closed subscheme and $f\:X'\to X$ is a $Z$-admissible blowing up with center $V\into Z$, and let $Z'$ and $E'$ be the transforms. Also, let $\tilZ=Z\times_X\tilX$ and $\tilE=E\times_X\tilX$, and let $\tilZ'$ and $\tilE'$ be their transforms under the blowing up of $\tilX$ along $V$. Then, $E'$ is an snc divisor in $X'$, $\tilZ'=\tilZ\times_{X'}\tilX'$ and $\tilE'=E'\times_{X'}\tilX'$.
\end{lem}
\begin{proof}
Recall that by Lemma \ref{blowuplem} $\tilf\:\tilX'\to\tilX$ is the blowing up along $V$. In particular, $E_f=V\times_XX'$ and $E_\tilf=V\times_\tilX\tilX'$ and it follows that $E_\tilf=V\times_X\tilX'=E_f\times_{X'}\tilX'$. Using that $(Z\times_XX')\times_{X'}\tilX'=Z\times_X\tilX'=\tilZ\times_{\tilX}\tilX'$, we obtain that also $\tilZ'=\tilZ\times_{X'}\tilX'$.

For concreteness assume that $E=\sum_{i=1}^nE_i$ is ordered and let $E'=\sum_{i=1}^{n+1}E'_i$ and $\tilE'=\sum_{i=1}^{n+1}\tilE'_i$. The above paragraph easily implies that $\tilE'_i=\tilE_i\times_{X'}\tilX'$ for any $i$. Since by the usual theory $\tilE'$ is an snc divisor with irreducible components $\tilE'_i$, it follows that $E'$ is snc.
\end{proof}

\subsubsection{Principalization}
Assume that $(X,E)$ is a ptm with a boundary and $Z\into X$ is a closed subscheme, then a {\em principalization} of $Z$ (or of its associated ideal $\calI_Z$) on $(X,E)$ is a sequence of $Z_i$-admissible blowings up $X_{i+1}=\Bl_{X_i}(V_i)\stackrel{f_i}\to X_i$ with $0\le i\le n-1$, where $(X_0,E_0,Z_0)=(X,E,Z)$ and for $0\le i<n$ one defines inductively $E_{i+1}$ and $Z_{i+1}$ to be the transforms of $E_i$ and $Z_i$, and $Z_n=\emptyset$. We will use dashed arrows to denote sequences of morphisms, e.g. $X_n\dashto X$.

\subsubsection{The principalization theorem}
It turns out that the usual principalization on $\tilX$ induces a principalization on $X$ yielding the following result:

\begin{theor}\label{princth}
There exists a method which to any triple $(X,E,Z)$ consisting of a ptm $X$ of finite type over a field $k$ of characteristic zero, a boundary $E$ and a closed subscheme $Z\into X$ associates a principalization $$\calP(X,E,Z)\:X_n\dashto X_0=X$$ so that $\calP$ depends on the triple $(X,E,Z)$ smooth-functorially: if $X'\to X$ is a smooth morphism of ptm's, $E'=E\times_XX'$ and $Z'=Z\times_XX'$, then $\calP(X',E',Z')$ is obtained by pulling back the blowings up sequence $\calP(X,E,Z)$ and removing all trivial blowings up (those with empty centers).
\end{theor}
\begin{proof}
Choose any smooth-functorial principalization method for varieties, e.g. the method from \cite{Wlodarczyk}. Set $\tilE=E\times_X\tilX$ and $\tilZ=Z\times_X\tilX$ and let $\tilX_{i+1}=\Bl_{V_i}(\tilX_i)$, $0\le i<n$ be the principalization of $\tilZ$ on $(\tilX,\tilE)$. We pushforward the blowing up sequence $\tilX_n\dashto\tilX_0=\tilX$ by setting $X_0=X$ and $X_{i+1}=\Bl_{V_i}(X_i)$. This makes sense because applying Lemma~\ref{blowuplem} inductively we obtain that $\tilX_i$ is, indeed, the reduction of $X_i$ and hence $V_i\into\tilX_i\into X_i$. Applying Lemma~\ref{transformlem} inductively we obtain that $V_i$ has snc with the boundary $E_i$, which is the transform of $E_{i-1}$, and $V_i$ is a subscheme of $Z_i$, which is the transform of $Z_{i-1}$. In other words, the blowings up are $Z_i$-admissible. Finally, Lemma~\ref{transformlem} also implies that $\tilZ_i=Z_i\times_{X_i}\tilX_i$ for any $i\le n$. Since $\tilZ_n$ is empty this implies that $Z_n$ is empty, and hence the sequence $X_n\dashto X_0=X$ is a principalization of $Z$, that we denote $\calF(X,E,Z)$.

Smooth-functoriality of the constructed principalization follows from the smooth-functoriality of the usual principalization and the easily checked fact that all other ingredients, such as the reduction morphism $\tilX\to X$, are compatible with smooth base changes.
\end{proof}

\subsection{Divisors with monomial reduction}\label{monomsec}

\subsubsection{Modifications with a trivial reduction}
Let $f\:Y\to X$ be a morphism of ptm's. Then the following conditions are equivalent: (i) $f$ is a bijective modification, (ii) is a modification and a homeomorphism, (iii) $f$ is generically an isomorphism and the reduction $\tilf$ is an isomorphism. If these conditions are satisfied, we say that $f$ is a {\em modification with a trivial reduction}.

\begin{exam}
An important example of a modification with a trivial reduction is the blowing up $Y=\Bl_\tilD(X)$ along a divisor $\tilD\into\tilX$ of the reduction. Locally on $X$ we can assume that $X=\Spec(A)$ is affine and $f\in A$ is such that its image $\tilf\in A/\calN$ defines $\tilD$, where $\calN$ is the nilradical. Then $\tilD=V(\calN+(f))$ in $X$ and $Y=\Spec(A[\frac{\calN}{f}])$ consists of the $f$-chart, since other charts correspond to nilpotent elements and hence are empty.
\end{exam}

More generally, if $\tilD_1\.\tilD_l$ are divisors of $\tilX$ then by induction on $l$ one can define a sequence $X_l\dashto X_0=X$ with $X_{i+1}=\Bl_{\tilD_i}(X_i)$ because $\tilD_i\subset\tilX=\tilX_i$.

\subsubsection{Monomialization}
We saw in Remark \ref{sncrem} that if $D$ is a divisor with an $\tilE$-monomial reduction, $D$ itself does not have to be monomial with respect to any boundary or it can be monomial with respect to another boundary. All these problems are remedied by applying a simple blowing up sequence.

\begin{lem}\label{monomiallem}
Assume that $X$ is a ptm with an ordered boundary $E$ and $D$ is a Cartier divisor such that $\tilD$ is $\tilE$-monomial. Let $\tilD=\sum_{i,n}n\tilD_{i,n}$ be the decomposition, where $\tilD_{i,n}$ is the union of components of $D$ of multiplicity $n$ that lie in $\tilE_i$. Let $X'\dashto X$ be the modification with trivial reduction obtained by blowing up centers $V_{1,1}\.V_{1,n_1},V_{2,1},\dots$, where $V_{i,j}=\coprod_{n\ge j}\tilD_{i,n}$. Then $D\times_XX'$ is monomial with respect to the transform $E'$ of $E$.
\end{lem}
\begin{proof}
It suffices to check the claim locally at a point $x\in X$. Let $n_i\ge 0$ be such that $x\in \tilD_{i,n_i}$. Choose parameters $\veps,t_1\. t_l$ so that $E=V(t_1\dots t_r)$ at $x$. Then locally at $x$ we have that $\tilD=V(\prod_{i=1}^r\tilt_i^{n_i})$ and hence $D=V(f)$ for $f=\prod_{i=1}^rt_i^{n_i}+\veps a$. Since $\veps'=\veps/\prod_{i=1}^rt_i^{n_i},t_1\.t_n$ are parameters of $X'$ at $x$ we obtain that $f=\prod_{i=1}^rt_i^{n_i}(1+\veps' a)$ in $\calO_{X',x}$ and $1+\veps' a$ is a unit. This finishes the proof.
\end{proof}

For the sake of completeness we note that the same sequence is produced by the usual principalization:

\begin{rem}
In fact, the sequence $\tilX'\dashto\tilX$ in the above lemma is precisely the principalization blowings up sequence of $\tilD$ on $(\tilX,\tilE)$ produced by the classical order reduction algorithms with marking 1. So, similarly to the argument in the proof of Theorem~\ref{princth}, its pushforward $X'\dashto X$ in the lemma is a principalization of $D$.
\end{rem}

\subsection{Resolution of $B$-schemes}\label{Bschemes}

\subsubsection{Distinguished $B$-pairs}
Let $B=\Spec(k[\pi]/(\pi^n))$. By a distinguished $B$-pair we mean a morphism $f\:X\to B$ and a divisor $E$ on $X$ such that $X$ is a ptm, $E$ is a boundary on $X$, and $\pi\calO_{X}=\calN\calI$, where $\calN$ is the nilradical of $\calO_{X}$ and $V(\calI)$ is an $E$-monomial divisor. In down to earth terms this means that locally on $X$ there exists coordinates $(\veps,t_1\.t_n)$ such that $E=V(t_1\.t_r)$ and $f^\#(\pi)=\veps t_1^{d_1}\.t_r^{d_r}$. In particular, $X\to B$ locally factors through a morphism $h\:X\to X_0=\Spec(k[\veps,t_1\.t_n]/(\veps^n))$, and if $X\to B$ is of finite type, then $h$ is \'etale.

\subsubsection{The main result}
Now we can prove our main result on resolution of $B$-schemes.

\begin{theor}\label{mainth1}
Let $k$ be a field of characteristic zero and $B=\Spec(k[\pi]/(\pi^n))$. Then there exists a procedure $\calF$ which obtains as an input a generically smooth morphism of finite type $X\to B$ and a closed nowhere dense subscheme $Z\into X$, and outputs a modification $\calF(X,Z)\:X'\to X$ with a $X'$ a ptm and a boundary $E'$ on $X'$ such that $Z'=Z\times_XX'$ is an $E'$-monomial divisor and $(X',E')$ is a distinguished $B$-pair. In addition, $\calF$ depends smooth functorially on the input: if $Y\to X$ is smooth and $T=Z\times_XY$, then $\calF(Y,T)=\calF(X,Z)\times_XY$.
\end{theor}
\begin{proof}
First, by Theorem~\ref{nonembeddedth} there exists a modification $X_1\to X$ with $X_1$ a ptm. Second, applying Theorem~\ref{princth} to $X_1$ and $Z_1\times_XX_1$ we obtain a modification of ptm's $X_2\to X_1$ and an ordered boundary $E_2$ on $X_2$ such that $Z_2=Z\times_XX_2$ is $E_2$-monomial. Third, the generic smoothness of $f$ implies that generically $\pi$ generates the nilradical $\calN_2\subset\calO_{X_2}$ on an open subscheme $U$. Principalizing $X_2\setminus U$ on $(X_2,E_2)$ by Theorem~\ref{princth} we obtain a modification $X_3\to X_2$, such that $X_3$ is provided with an ordered boundary $E_3$ and $\calN_3|_U=(\pi)|_U$ for $U=X_3\setminus E_3$.

Now, locally on $X_3$ there exists a factorization $(\pi)=\calN_3\calI$, and the locally principal ideal $\calI$ is invertible because it is trivial at the generic point and $X_3$ has no embedded components. Note that $\calI$ does not have to be $E_3$-monomial, moreover it is not uniquely defined hence the factorization does not have to globalize. However, blowing up the components of $E_3$ as in Lemma~\ref{monomiallem} we obtain a modification $X'\to X_3$ with a trivial reduction such that the pullbacks of each $\calI$ as above become monomial, that is, on $X'$ we locally have factorizations $(\pi)=\calN'\calI$ with an $E'$-monomial $\calI$. Such a factorization is unique because if $(\pi)=\calN'\calI'$ is another factorization with an $E_3$-monomial $\calI'$, then $V(\calI)$ and $V(\calI')$ restrict to the same divisor on $\tilX_3$ and hence  contain the same components of $E_3$ with the same multiplicities. Thus the factorization $(\pi)=\calN\calI$ globalizes, and this shows that $X'\to X$ is as required.

The smooth-functoriality of the construction follows from functoriality of all ingredients we used.
\end{proof}

\subsection{Other geometric spaces}\label{othersec}
We deduced our results from the classical principalization theorem for algebraic varieties. Now, let us discuss what can be done for more general schemes, complex analytic spaces, etc. These are nowadays standard reductions that were described in details in a few papers, so we only outline them.

\subsubsection{Analytic spaces}\label{spacesec}
It is folklore knowledge that the classical principalization theorem and its proof apply almost verbatim for complex analytic spaces and non-archimedean analytic spaces of characteristic zero, though, to the best of our knowledge, the only formalized framework, in which this was checked, is given in \cite{Bierstone-Milman}. Starting with principalization for analytic spaces one can establish analogues of Theorems~\ref{nonembeddedth}, \ref{princth} and \ref{mainth} using precisely the same arguments as in the paper.

\subsubsection{Schemes with enough derivations}
In fact, the classical principalization algorithm and its justification applies to the wider context of ideals on regular schemes with enough derivations, where the latter were defined in \cite[Remark~1.3.2(iii)]{non-embedded}. The only new (and simple) thing one should check is that any smooth blow up of an $X$ with enough derivations also has enough derivations. Strictly speaking, this was also never published, but a much more involved case of an algorithm in the setting of enough derivations was worked out with all details in \cite{ATW}. Moreover, the functoriality holds with respect to arbitrary regular morphisms and not only the smooth ones. Once principalization for such schemes is known, our arguments extend to this setting verbatim, and one obtains that Theorems~\ref{nonembeddedth}, \ref{princth} and \ref{mainth} hold for schemes locally embeddable into regular schemes with enough derivations, and the functoriality is with respect to arbitrary regular morphisms. Then the usual formal argument (see \cite[\S5.2]{non-embedded}) implies that the same results extend to analytic spaces (recovering the claim of \S\ref{spacesec}) and formal schemes locally embeddable into regular formal schemes with enough derivations.

\subsubsection{Arbitrary qe schemes}
It seems certain that the maximal generality, in which Theorems~\ref{nonembeddedth}, \ref{princth} and \ref{mainth} should hold, is that of arbitrary qe schemes of characteristic zero. However, resolution of non-reduced schemes was not established in such generality and only a weaker form of principalization was proved in \cite[Theorem~1.1.11]{Temkin-embedded} -- it outputs a regular scheme with a monomial ideal, but the intermediate steps can be singular. This would suffice to deduce the last two theorems, but it cannot be used for the non-embedded resolution of generically nil-principal schemes. All in all, the only stumbling block is resolution of arbitrary non-reduced qe schemes.

\section{Complements}\label{mainsec}
In this section we will slightly refine our main result by constructing a global embedding of a modification into a log smooth $B$-scheme. This requires an additional preparation.

\subsection{Factorization}\label{factorsec}
In addition to the analogues of the usual resolution and principalization we will need certain results about factorization and resolution of indeterminacies for ptm's. We will only cover the cases that will be needed in our applications.

\begin{theor}\label{factorth}
Assume that $f\:Y\to X$ is a modification of ptm's with a trivial reduction. Assume, in addition, that there exist an ordered boundary $E=\sum_{i=1}^lE_i$ on $X$ such that $f$ is an isomorphism over $X\setminus E$. Then there exists a sequence of blowings up of components of $\tilE_i$'s $Y'\dashto Y$ such that the composed morphism $f'\:Y'\to X$ splits into a composition of blowings up of components of $\tilE_i$'s. The sequence $Y'\dashto Y$ and the splitting of $f'$ depend on the datum functorially with respect to smooth morphisms $\oX\to X$ and pullbacks $\of\:\oY=Y\times_X\oX\to\oX$ and $\oE=E\times_X\oX$.
\end{theor}
\begin{proof}
To simplify notation we identify the underlying topological spaces of $X$ and $Y$. Notice that locally we can factor $\calN_X\calO_Y$ as $\calI\calN_Y$, where $\calI$ is invertible -- just take local generators $\veps_Y$ and $\veps_X$, then $\veps_X=a\veps_Y$ and we take $\calI=(a)$. This does not yield a global divisor $\calN_Y/\calN_X$ because an element $a$ is defined only up to adding an element annihilated by $\veps_Y$, but the reduction $(\tila)$ is unique and we obtain a global divisor on $\tilX$ that we denote $\tilD=\wt{\calN_Y/\calN_X}$. Clearly, it is $\tilE$-monomoial, so we fix the decomposition $\tilD=\sum_{i,n}n\tilD_{i,n}$ with $\tilD_{i,n}\subset\tilE_i$. Now, let $Y'\dashto Y$ be the sequence obtained by successive blowing up the centers $V_{i,j}=\coprod_{n\ge j}\tilD_{i,n}$ and let $E'$ be the transform of the boundary to $Y'$. By Lemma~\ref{monomiallem} the local divisors $(a)$ become $E'$-monomial on $Y'$. The reductions $(\tila)$ agree on the intersections (this happens already on $X$), hence we obtain a factorization $\calN_{Y'}=D'\calN_X$ with an $E'$-monomial divisor $D'$.

Choose the presentation $D'=\sum_{i,n}nD'_{i,n}$ with $D'_{i,n}\subset E_i$ (we do not need this, but in fact only even $n$'s show up and $\tilD'_{i,2n}=\tilD_{i,n}$). We claim that $Y'\to X$ splits into the blowing up sequence $Y'=X_m\dashto X_0=X$ with centers $V'_{i,j}=\coprod_{n\ge j}\tilD'_{i,n}$. Indeed, this can be checked locally at a point $x\in X$. Choose parameters $\veps_X,t_1\.t_n\in\calO_{X,x}$ such that $E=V(t_1\dots t_r)$ and let $n_i$ be such that $x\in D'_{i,n_i}$. Then $\veps_Y=\veps_X/\prod_{i=1}^rt_i^{n_i}$ is a generator of the nilradical of $\calO_{Y',x}$ and hence an embedding $\calO_{X_m,x}\subseteq\calO_{Y',x}$ arises. Moreover, it is an isomorphism by Nakayama's lemma because the reductions coincide and $\calO_{X_m,x}$ contains a generator of the nilradical of $\calO_{y',x}$. Thus, $Y'=X_m$ and we have constructed a splitting of $Y'\to X$. The check that our construction is smooth-functorial is straightforward and we omit it.
\end{proof}

\begin{cor}\label{morphismCor}
	Let $X\rightarrow B_n=\Spec(k[\pi]/(\pi^n))$ be generically smooth. Then there exists a construction $\mathcal{G}(X\rightarrow B_n)=Y\rightarrow X\rightarrow B_n$ such that the composed morphism is smooth away from an snc divisor. This construction is functorial with respect to smooth morphism.
\end{cor}
\begin{proof}
	As $X\rightarrow B_n$ is generically smooth, $X$ is generically nilprincipal and we may apply Theorem \ref{nonembeddedth} to get $X'\rightarrow X$ such that $X$ is a ptm. Let $Z\subset X$ be the singular locus of the composed morphism $X'\rightarrow X\rightarrow B_n$, and by Theorem \ref{princth} we have a principalization $\mathcal{P}(X', \emptyset, Z):Y\rightarrow X'$ such that the composed morphism $Y\rightarrow B_n$ is smooth away from an snc divisor. This process is smooth-functorial as both ingredients are.
\end{proof}
\begin{rem}\label{etaleloc}
	We make two remarks on this result:
	\begin{enumerate}
		\item The morphism $Y\rightarrow X'$ is a modification of ptms with trivial reduction, so if we wish to apply Theorem \ref{factorth} we may.
		\item In practice, this means that given a generically smooth $X\rightarrow B_n$ there is a resolution which \'etale-locally looks like $X'=\text{Spec}k[\epsilon, t_1,..., t_l]/(\epsilon^n)\rightarrow B_n$ given by $\pi\mapsto\epsilon\cdot t_1^{n_1}...t_l^{a_l}$.
	\end{enumerate}
\end{rem}

\subsection{Retracts}\label{retractsec}
By a retract of a ptm $X$ we mean a morphism $r\:X\to\tilX$ which is a retract of the closed immersion $i\:\tilX\into X$, that is, $r\circ i=\id_\tilX$. In general, a retract does not exist, but if $X$ is of finite type over a field $k$, then it exists whenever $X$ is affine. Indeed, $X$ is a nilpotent thickening of $\tilX$ and $\tilX$ is smooth over $k$, hence the $k$-morphism $\id_\tilX$ lifts to a $k$-morphism $r\:X\to\tilX$. (For an arbitrary ptm $X$, a retract at least exists at the generic points $\eta\in X$, namely, $r_\eta\:\eta\to\tileta$ corresponds to a choice of a field of definition $k(\tileta)\into\calO_{X,\eta}$ of the Artin ring $\calO_{X,\eta}$.) We will show that after blowing up $X$ one can extend a generic retract to a retract of the whole $X$. The argument will be similar to the proof of Theorem~\ref{factorth} but a bit more elaborate. Here is the main particular case we will need.

\begin{lem}\label{retlem}
Let $X$ be a ptm of finite type over a field $k$ with nilradical $\calN\subset\calO_X$, let $D=V(\calI)$ be a Cartier divisor with a smooth reduction $\tilD=D\times_X\tilX$ and complement $X_0=X\setminus D$, and let $i\:X_0\into X$ denote the open immersion. Assume that $D$ is irreducible with generic point $\nu$ and $r_0\:X_0\to\tilX_0$ is a retract of $X_0$, then

(i) The retract $r_0$ extends to $r$ if and only if the homomorphism $\phi\:i_*\calO_{\tilX_0}\into i_*\calO_{X_0}$ induced by $r_0$ takes $\calO_\tilX$ to $\calO_X$.

(ii) The following three conditions are equivalent and they hold for a large enough $n$:
(a) $\phi$ takes $\calO_\tilX$ to $\calO_X[\calI^{-n}\calN]$,
(b) $\phi$ takes $\calO_{\tilX,\nu}$ to $\calO_{X,\nu}[\calI_\nu^{-n}\calN_\nu]$,
(c) $r_0$ extends to a global retract on $\Bl_{\widetilde{nD}}(X)$.

(iii) Assume that $n\ge 1$ is such that the conditions in (ii) are satisfied and let $X'=\Bl_\tilD(X)$. Then $\phi$ takes $\calO_{\tilX'}$ to $\calI^{1-n}\calO_{X'}$.
\end{lem}
\begin{proof}
All claims can be checked locally on $X$, where we use in (ii) that $X$ is quasi-compact. Thus, we can assume that $X=\Spec(A)$ and $\calI=(t)$, and then $X_0=\Spec(A_t)$, $\tilX=\Spec(\tilA)$ and $\tilX_0=\Spec(\tilA_t)$, where $\tilA=A/\calN$. Let $s_0\:\tilA_t\to A_t$ be the section corresponding to $r_0$. Since $\tilX\into X$ is a homeomorphism, $r_0$ extends to $r$ if and only if the homomorphism of sheaves extends, that is $s_0(\tilA)\subseteq A$. This proves (i).

Recall that $X'=\Bl_{\widetilde{nD}}(X)=\Bl_{(t^n)+\calN}(X)$ consists of the $t$-chart, hence $X'=\Spec(A_n)$, where $A_n=A[t^{-n}\calN]$. Clearly, $X'\to X$ is a homeomorphism and we denote the preimage of $\nu$ by $\nu'$. Applying (i) to $X'$ we obtain that (a) and (c) are equivalent, and the equivalence of (a) and (b) reduces to the claim that if $r_0$ extends to $\nu'$, then $r_0$ extends to the whole $X'$. But $X'$ can be obtained from the ptm $X$ by $n$ successive blowings up of $\tilD$, hence $X'$ is a ptm and any regular function on $X_0$ that extends to $\nu'$ automatically extends to the whole $X'$ by Corollary~\ref{CMcor}. The claim about extension of $r_0$ follows.

Let us now prove that for a large $n$ the condition (a) is satisfied. Given $\tila\in\tilA$ choose a lift $a\in A$ and notice that $s_0(\tila)-a\in\calN_t$ and hence lies in a subring $A_n=A[t^{-n}\calN]$ of $A_t$. Choosing a set $\tila_1\.\tila_l$ of $k$-generators of $\tilA$ and a large enough $n$, we obtain that $s_0(\tila_i)\in A_n$ for any $1\le i\le l$ and hence $s_0(\tilA)\subset A_n$. This proves the second part of (ii).

Finally, in (iii) $X'=\Spec(A')$, where $A'=A_1$, and hence the nilradical of $A'$ is $\calN'=t^{-1}\calN$ and $\tilA=\tilA'$. Therefore, $s_0(\tilA')\subset A_n=A'[t^{1-n}\calN']$, proving (iii).
\end{proof}

\begin{theor}\label{retractth}
There exists a method which to any ptm with a boundary $(X,E)$ of finite type over a field $k$ of characteristic zero and a generic retract $r_\eta\:\eta\to\tileta$, where $\eta=\coprod_{i=i}^n\Spec(\calO_{X,\eta_i})$ and $\eta_1\.\eta_n$ is the set of generic points of $X$, associates a sequence of admissible blowings up $$\calR(X,E,r_\eta)\:(X',E')\dashto (X,E)$$ such that $r_\eta$ extends to a retract $r\:X'\to\tilX'$. Moreover, one can construct $\calR$ functorially with respect to \'etale morphisms $Y\to X$ provided with a compatible pair of generic retracts $r_{\eta_Y}$ and $r_{\eta_X}$.
\end{theor}
\begin{proof}
We will first construct the retract as a composition of two admissible blowings up sequences, and only then discuss the functoriality. To simplify notation we assume that $X$ is irreducible and $\eta$ is the generic point, but the argument works in general as well. Let $s_\eta\:k(\eta)\into\calO_{X,\eta}$ be the field of coefficients corresponding to $r_\eta$. Recall that $X$ has no embedded components, hence $\calO_{X,x}\subseteq\calO_{X,\eta}$ for any $x\in X$ and by Lemma~\ref{retlem}(i) $r_\eta$ extends to $r$ locally at $x\in X$ if and only if $s_\eta(\calO_{\tilX,x})\subseteq\calO_{X,x}$. Clearly, this is an open condition and we denote by $U$ the open set of such points $x\in X$, and denote by $Z=X\setminus U$ the complement. In other words, $Z$ is the indeterminacy locus of the rational map $r_\eta\:X\to\tilX$. The first part of $\calR$ is the principalization $\calP(X,E,Z)\:X'=X_l\dashto X_0=X$ of $Z$. Let $E'_1\.E'_l$ be the components of $E'$ created by $\calP$, then the indeterminacy locus $Z'$ of $r_\eta\:X'\to\tilX'$ is contained in $E''=\sum_{i=1}^lE'_i$.

The remaining procedure iteratively blows up unions of components of $\tilE''$. In particular, the reduction $\tilX'$ remains unchanged, so we can identify $\tilE'_i$ with subschemes of further blowings up. The choice of the center on each step is done by the following rule: for any generic point $\nu\in E''$ let $i=i(\nu)$ be the number for which $\nu\in E'_i$ and let $n(\nu)$ be the minimal $n\ge 0$ which satisfies the condition of Lemma~\ref{retlem}(ii). The step takes all generic points $\nu$ such that $n(\nu)>0$ (i.e. the point is in the indeterminacy locus) and the lexicographic value of $(n(\nu),i(\nu))$ is maximal, and blows up the reduced closed subschemes with these generic points. In other words, we blow up components of $\tilE'_i$ with the maximal value of the invariant $(n(\nu),i(\nu))$. The maximal value of the invariant drops after each blowing up by Lemma~\ref{retlem}(ii) and (iii). Part (i) of the same lemma says that once $n$ drops to zero, the indeterminacy locus is empty, that is, $r_\eta$ extends to a global retract.

Finally, let us discuss the functoriality. So let us assume that $f\:Y\to X$ is \'etale, $E_Y=E\times_XY$ and a generic retract $r_{\eta_Y}$ is the pullback of $r_{\eta_X}$. The principalization is functorial with respect to smooth morphisms, so we should only check that the indeterminacy locus and the invariant $n(\nu)$ used in the second step are compatible with $f$. Moreover, if $D$ is the irreducible component of $\nu$, then by Lemma~\ref{retlem}(ii) $n(\nu)$ is the minimal $n$ such that blowing up $\widetilde{nD}$ resolves the indeterminacy at $\nu$. Blowings up are compatible with smooth (even flat) morphisms, hence it suffices to check functoriality of the indeterminacy locus: if $y\in Y$ and $x=f(y)$, then $r_{\eta_X}$ extends to $x$ if and only if $r_{\eta_Y}$ extends to $y$. Furthermore, since the indeterminacy locus is open, localizing $X$ and $Y$ appropriately we can reduce the problem to the following particular case: $r_{X_\eta}$ extends to a global retract $r_X$ if and only if $r_{Y_\eta}$ extends to a global retract $r_Y$. Clearly, it suffices to check the latter locally, so we can assume that $X$ and $Y$ are affine.

Assume that $r_{\eta_X}\:\eta_X\to\eta_\tilX$ extends to $r\:X\to\tilX$. Then a morphism $Y\to X\to\tilX$ arises and the $\tilX$-morphism $\id_\tilY$ lifts uniquely to a $\tilX$-morphism $r_Y\:Y\to\tilY$ by the \'etalenes of $\tilY$ over $\tilX$ and affineness of $Y$. Thus, the retract $r_X$ lifts uniquely to a retract $r_Y$. Since the lift $r_{\eta_Y}$ of $r_{\eta_X}$ is also unique by the same argument, both agree, that is $r_Y$ is the extension of $r_{\eta_Y}$. Conversely, if $r_{\eta_Y}$ extends to $r_Y$, then it is easy to see that $r_{\eta_X}$ extends to $r_X$.

\end{proof}

We do not know if functoriality holds for smooth morphisms as well.

\subsection{Log blowings ups}\label{logblowsec}
We will need one more construction on a ptm with boundary. Informally speaking, it replaces boundary components by varieties intersecting $X$ so that one obtains a thick version of a normal crossings variety. The most convenient way to realize this is by using log geometry.

\subsubsection{Associated log structure}
As often happens one can encode the boundary in a log structure, and in various applications this is even a more conceptual approach. Given a ptm with a boundary $(X,E)$ we define the associated log structure $M_X$ as follows: for a point $x\in X$ let $t_0=\veps$ be a generator of the nilradical of $\calO_x$ and let $t_1\.t_r\in\calO_x$ be parameters that define the irreducible components of $E$ passing through $x$ and consider the prelogarithmic structure $\oM_x=\oplus_{i=0}^r\bbN\log(t_i)\to\calO_x$ which sends $\log(t_i)$ to $t_i$. Since the choice of $\veps$ and $t_i$ is unique up to a unit, the associated log structure $M_x$ is independent of the choices, and it is easy to see that the local definitions patch to a global log structure $M_X$ on $X$. Clearly, $E$ can be reconstructed from $M_X$, so we also address to the log scheme $(X,M_X)$ as a {\em ptm with a boundary}.

\subsubsection{Log blowings up of reduced divisors}
We will use a very special form of log blowings up of a ptm with a boundary $(X,M_X)$.

\begin{lem}\label{logblowlem}
Let $(X,M_X)$ be a ptm with a boundary and $\tilD$ a smooth component of the induced boundary $\tilE$ of $\tilX$. Then $X''=\Bl_\tilD(X)$ is an irreducible component of $X'=\LogBl_\tilD(X)$. In addition, $X''$ is a monoidal subscheme of $X'$.
\end{lem}
\begin{proof}
The claim can be checked locally over a point $x\in\tilD$, so assume that $X=\Spec(A)$ and $(\veps=t_0,t=t_1\.t_n)$ is a family of parameters such that $\tilD=(\veps,t)$ and $E=V(t_1\dots t_r)$ at $x$. Then $Y=\Spec(\bbZ[t_0\.t_r])$ is a chart for $X$, and hence $X'\to X$ is the base change of the log blowing up of $Y$ along $(t_0,t_1)$. Since $Y$ is log regular, the latter is the usual blowing up described by the charts, and hence $X'$ is glued from two charts: $X'_t=\Spec(A[\veps']/(t\veps'-\veps))$ and $X'_\veps=\Spec(A[t']/(t'\veps-t))$.

Recall that $X''=\Spec(A'')$, where $A''=A[\frac{\veps}{t}])$, and note that $A''$ is the image of $A'=A[\veps']/(t\veps'-\veps)$ in $A_t$, hence $X''$ is the irreducible component of $X'$, is the strict transform of $X$ and lies in the $t$-chart. Furthermore, it is easy to see that $A''=A'/(\veps'^h)$, where $h$ is the thickness of $X$, hence $X''$ is given by vanishing of the monomial ideal $(\veps'^h)$.
\end{proof}

Here is a concrete typical example.

\begin{exam}
Let $A=k[t,\veps]/(\veps^n)$ and $X=\Spec(A)$ with the log structure induced by $\bbN\log(\veps)\oplus\bbN\log(t)$. Then $\tilD=V(t,\veps)$ is a monomial subscheme, which is also a smooth divisor in $\tilX$. The log blowing up $X'=\LogBl_\tilD(X)$ is glued from two charts: $$X'_t=\Spec(A[\veps']/(t\veps'-\veps))=\Spec(k[\veps',t]/(t^2\veps'^2))$$ and $$X'_\veps=\Spec(A[t']/(t'\veps-t))=\Spec(k[t',\veps]/\veps^2).$$ Thus, $X'$ has two irreducible components. The first component is $\Bl_\tilD(X)=\Spec(k[t,\veps'])/(\veps'^2)$; it is contained in $X'_t$. The other component is a ptm with nilradical $(\veps)$ and reduction isomorphic to $\bfP^1_k$ with coordinate $t'$; it is contracted to $\tilD$ in $X$.
\end{exam}

For our applications we will need to construct sequences of log blowings. The following results follows from Lemma~\ref{logblowlem} by an obvious induction on the length of the sequence.

\begin{cor}\label{logblowcor}
Let $(X,M_X)$ be a log ptm with a boundary, and let $\tilD_1\.\tilD_l$ be smooth components of the associated snc boundary $\tilE\subset\tilX$. Then there exists a sequence of log blowings up $X_l\dashto X_0=(X,M_X)$ such that the following conditions are satisfied: the strict transform $X'_i$ of $X$ in $X_i$ is a monomial subscheme of $X_i$, each $X'_{i+1}$ is the blowing up of $X'_i$ along $\tilD_i$, in particular, each $X'_i\to X$ is a modification with the trivial reduction and $\tilD_i\into X'_i\into X_i$ are monomial subschemes, and $X_{i+1}=\LogBl_{\tilD_i}(X_i)$.
\end{cor}

\subsection{Log smooth resolution}
To get our last main result it remains to assembly various pieces we have developed altogether.

\begin{theor}\label{mainth}
Assume that $k$ is a field of characteristic zero, $B=\Spec(k[\pi]/(\pi^n))$ and $X\to B$ is a generically smooth morphism of finite type with a closed nowhere dense subscheme $Z\into X$. Let $M_B$ be the log structure on $B$ generated by $\bbN\log(\pi)\to M_B$, where $\log(\pi)$ is mapped to $\pi$. Then there exists a log smooth morphism $(Y,M_Y)\to(B,M_B)$, an irreducible component $X'$ of $Y$ and modification $X'\to X$, such that $Z'=Z\times_XX'$ is a divisor and both $X'$ and $Z'$ are monomial subschemes of $Y$. In addition, the choice of $Y$ and $X'$ can be made depending \'etale-functorially only on $X\to B$, $Z\into X$ and a generic retract $r_\eta\:\eta_X\to\eta_\tilX$, and if $X\to B$ is proper one can also choose $Y\to B$ to be proper.
\end{theor}
\begin{proof}
Acting as in the proof of Theorem~\ref{mainth1} we can find a modification $X_1\to X$ such that $X_1$ is a ptm with an ordered boundary $E_1$ and $Z_1=Z\times_XX_1$ is $E_1$-monomial. Next, the generic smoothness of $f$ implies that generically $\pi$ generates the nilradical $\calN\subset\calO_{X_1}$ on an open subscheme $U$. Principalizing $X_2\setminus U$ on $(X_2,E_2)$ by Theorem~\ref{princth} we obtain a modification $X_2\to X_1$, such that $\calN|_U=(\pi)|_U$ for $U=X_2\setminus E_2$. Next, we apply Theorem~\ref{retractth} to construct a modification $g\:X'\to X_2$ with a ptm $X'$ and an ordered boundary $E'$ containing the preimage of $E_2$ such that $X'$ possesses a retract $X'\to\tilX'$.

Set $U'=X'\setminus E'$ and $Y'=\tilX'\times_{\tilB}B=\Spec_{\tilX'}\calO_\tilX'[\pi]/(\pi^n)$. Then the natural morphism $f\:X'\to Y'$ is a modification with a trivial reduction. In addition, $f$ is an isomorphism over the complement to $\tilE'$, hence by Theorem~\ref{factorth} replacing $X'$ once again by a successive blowing up along components of $E'$, we can achieve that $f$ factors into a composition $X'=Y'_n\dashto Y'_0=Y'$ of blowings up of components of $\tilE'_i$'s, say $Y'_{i+1}=\Bl_{V_i}(Y'_i)$.

Finally, we provide $Y'$ with the log structure combined from the log structures induced from $B$ and the log structure on $\tilY'=\tilX'$ given by $\tilE'$. More formally, consider the closed immersion $i\:\tilU'\into\tilY'$, then $M_{\tilY'}=i_*\calO_{\tilU'}^\times\cap\calO_{\tilY'}$ provides $\tilX$ with the structure of a log smooth log scheme over $k$ (with the trivial log structure $k^\times\into k$) and we set $(Y',M_{Y'})=(\tilY',M_{\tilY'})\times_{\Spec(k)}(B,M_B)$. In particular, the projection $(Y,M_{Y'})\to(B,M_B)$ is log smooth. By Corollary~\ref{logblowcor} there exists a sequence of log blowings up $Y=Y_n\dashto Y_0=Y'$ such that $Y'_i\into Y_i$ is the irreducible component which is a modification of $Y'$ and the centers are log subschemes $V_i\into\tilY'=\tilY'_i\into Y'_i\into Y_i$. Thus, $X'$ is an irreducible component of $Y$ and since the log blowings up are log \'etale the composition $Y\dashto Y'\to B$ is log smooth.

The \'etale-functoriality of the construction follows from functoriality of all ingredients we used.
\end{proof}

\bibliographystyle{amsalpha}

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Exploring the intersection of life, running and the business of online communities
Live Music Exploration: Chicago House, Daft Punk and The Fan Experience
February 5, 2011 ChrisUncategorized
Do you ever go to a concert, or hear someone talk about their favorite concert experience, and then find yourself diving back into the music to compare a classic live album against the concert stories of the past?
In the days since we went to the WBEZ Hip Hop Block Party, namely the Chicago house music tribute at the end, I've been thinking about how live house music and the fan experience has evolved over the years.
And as I've been exploring the roots and evolution of the live house music experience, I've been revisiting a lot of Frankie Knuckles's classic tracks like "Your Love" (iTunes) and wondering what it was like in a place like the Warehouse in Chcago in the late 70 and early 80's, and how those club experiences would compare to experiences captured on Daft Punk's 2007 Alive 2007 (Live) album (iTunes).
Before the WBEZ Winter Block Party, I already had a knowledge of Frankie Knuckles and his revolutionary impact on and virtual creation of house music, but it was during the Q&A and especially the righteous dance culture spoken word piece by Reggie Gibson that really sent me back to the legacy of Chicago house and its influence on groups like Daft Punk.
So as I've been driving around in my car, I've been absorbing and zeroing in on the way the crowd responds vocally to the music as I play Daft Punk's live album over and over again.
I Can't Get Enough of That Roaring Pleasure
What about those waves of roaring pleasure and groove that come surging from the crowd during Daft Punk's 2007 set in Paris?
Did fans at that concert feel the same rush of emotions and freedom that fans felt as Frankie Knuckles rocked the crowd at the Warehouse in Chicago?
Listening to the music, it's obvious that Daft Punk is celebrating, re-interpreting and innovating the influential Chicago house sound by muscling up the classic house sound with heavy doses of electro-rock, thumping bass and a killer crunch that rattles all parts of your soul.
And that experience, I imagine, would be similar to what I'm told it was like to experience a show at the Music Box when Ron Hardy was DJing.
What About the Collective and Individual Fan Experiences?
Considering the historical context of each house scene, and looking at things from a cultural and sociological perspective, it's safe to say that each a Draft Punk and Frankie Knuckles crowd were each gathering to "celebrate" for different reasons.
Some fans came to the Warehouse to just simply have fun and jack to themselves to oblivion, while others came to be a part of a bigger social movement driving by the music.
So I've been wondering if we were to compare the fans from both scenes, what would be the main differences between the Daft Punk crowd and the Warehouse crowd sociologically, emotionally and psychologically? And on the flip-side, what did those crowds have in common?
I've been letting my imagination run wild with all those thoughts and wonderings as I listen to Alive 2007's massive eruptions of sonic bliss pouring out from the crowd.
It's a prefect case study of crowd control to behold as the master duo of Thomas Bangalter and Guy-Manuel de Homem-Christo work the masses into a frenzy, taking fans higher and higher leading them through all the stages of sonic pleasure one fan could hope to experience at concert.
And as I consider this Daft Punk live album experience and compare it to what Reggie Gibson celebrates in his house music spoken word piece, I wonder what it would be like to listen to a recording of Frankie Knuckles rocking a Warehouse crowd.
What Role Did Drugs Play?
One of the things that I've continued to wonder about too, is the influence of drugs on both crowds.
During the WBEZ house music forum, they didn't talk to much about the role, if any, that drugs played in the Chicago dance club culture, and as we continue to explore the use of Ecstasy at rave and dance concerts, I wonder how different the crowds and the concert experience would be for both Daft Punk and Frankie Knuckles crowd, if the fans and the DJs were, or were not, influenced by the use of drugs. And what can learn by exploring a David Guetta live concert experience that's captured in the memories of the fans?
That said, there's an interesting quote, (at about the 2:25 mark in the video above), from house music documentary Pump Up the Volume. As Daft Punk's "One More Time" plays in the background the interviewee says "…[the early house scene] was about the music… it wasn't about poppin' E man!"
We'll get into that topic more as we continue our this and our ongoing community of groove exploration.
But, right now…
Now It's Your Turn
Tell us your favorite Warehouse or house music club stories. Tells us what you think are the differences or similarities between house and dance club culture now and then.
Spread the love...
ChicagoDaft PunkDancedance musicFrankie KnucklesHouseRon HardyThe Warehouse
For Your Downloading Pleasure: Super Bowl Flashback, Baeble Concert Video Round-up, Mixtapes and more!
UPDATE: Shure and Jansport Battle of the Bands Contests Rock On
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Stephanie King joined COPE Health Solutions as a Senior Consultant in December 2015. Prior to joining COPE Health Solutions, Ms. King served as Management Consultant for CHI St. Luke's Health – Baylor St. Luke's Medical Center in Houston, Texas. In her role as Management Consultant, Ms. King was responsible for successful development, implementation and reporting of the organization's two Texas 1115 Medicaid Waiver DSRIP projects, valued at over 20 million dollars. In addition to leading the organization's DSRIP efforts, Ms. King provided project management and operations analysis for several ambulatory care initiatives.
Prior to her work as Management Consultant, Ms. King served as Manager, Clinical Operations overseeing Baylor St. Luke's Medical Center's Kidney Transplant Clinic and ambulatory patient access team as well as HR Manager, Corporate Services, serving as the HR business partner for health system corporate departments. Before joining CHI St. Luke's, Ms. King served as Operations and Planning Coordinator at The Emory Clinic, the physician practice arm of Emory Healthcare in Atlanta, Georgia.
Ms. King holds a Master of Science in Healthcare Administration from The University of Alabama at Birmingham and a Bachelor of Science in Psychology and minor in Business from The University of Tennessee. Ms. King also serves as an Examiner for the Quality Texas Foundation, the organization that awards the Texas Award for Performance Excellence based on the Baldrige Criteria for Performance Excellence and is a member of the American College of Healthcare Executives. Outside of work, Ms. King enjoys spending time with family, running and traveling.


Whether you're moving from Overland Park to Colorado for work or family, you want the help of professional Long Distance Movers to help with your shift. Centurion Moving & Storage is a trusted Genuine Atlas Interstate Agent ready to assist with your Long Distance Movers!
Atlas Van Lines began over 70 years ago by relied on moving company entrepreneurs to develop a network of services serving customers better. There are guidelines and licenses required to be part of this moving network, something Centurion Moving & Storage is proud to hold & be a part of. We're likewise a BBB-accredited service along with a ProMover member of the American Moving & Storage Association with our Atlas Van Lines Partnership. How can we help you with your Long Distance Movers? Whether you're moving from Overland Park to Colorado for work or family, you desire the assistance of professional Long Distance Movers to assist with your shift. How can we assist you with your Long Distance Movers?
Selecting a premier Long Distance Movers-- one that can supply thorough service & help unrelatedly of the size and scope of your relocation-- is among the most important decisions you have to produce your moving. At Centurion Moving & Storage, we'& #39; ve concentrated on making our credibility one that is trusted & committed to our customers. As a counted on moving business, we provide our clients with a Move Concierge Coordinator, who will provide up-to-date interaction, ideas and help throughout your whole relocate to Colorado from Kansas City.
On top of using total moving services, Centurion Moving & Storage has storage easily available for your personal belongings-- both short- or long-term-- in our 50,000 square foot state-of-the-art warehousing center, conveniently situated in Kansas City. In addition to our owned center, we likewise have the Atlas network of storage centers situated in Colorado. We have alternatives for you if you desire your possessions saved in Overland Park or Colorado since of a gap in your moving-out and moving-in dates. If you are moving from the Overland Park area to Colorado while not being all set to receive their delivery, this is a practical alternative.
Call Centurion today to learn more about our Overland Park to Colorado Long Distance Movers capabilities. Or feel free to request a free moving estimate by filling out our online quote form.


>What happened in the US?
Everyone was ready for the Championship to be decided. All Hamilton had to do was come first which was not that unlikely given his unbeatable track record at the US (excuse the pun) and Vettel to come lower than second. However in the end Hamilton lost out to Kimi Raikkonen and Max Verstappen, clinching third (15 points) whilst Vettel came fourth (12 points).
The US Grand Prix was no victory for Vettel. He still finished behind Hamilton and 70 points now separate the two drivers. What does that mean? Vettel needs to win the final three races and Hamilton needs to come no higher than 10th in all of them. Hamilton to Win the championship is a safe bet.
I expect to see some Dutch magic from Verstappen again. This guy is unbelievable! He has always done very well in Mexico, last year he came first! I expect him on the podium this year if not win it again. Especially if there is rain, which is forecast.
Renault currently sits relatively comfortably in fourth thanks to noteworthy performances by Hulkenberg and Saniz over the year. Hulkenburg also has the pleasure of being best of the rest in the drivers championship, sitting behind the Mercedes, Ferrari and Red Bull Drivers.
However Sergio Perez is nipping at his heels, trailing by four points, and with Mexico being his home grand prix I expect he may take that spot.


If you or a loved one suffered a bleeding side effect from the use of Xarelto, including internal bleeding, brain hemorrhaging, blood clots or stroke, our attorneys may be able to help you to seek compensation for these injuries. To find out more about filing a Xarelto lawsuit, contact us today for a free case consultation.
Xarelto is a kind of blood thinner that belongs to a class of anticoagulants called direct factor Xa inhibitors. Anticoagulants, like Xarelto, interfere with the normal clotting of the blood. Clotting is necessary if a person is to stop bleeding from an open wound, but some people experience clotting when they are not injured.
In July 201, the FDA approved Xarelto (rivaroxaban), manufactured by Bayer and marketed by Janssen Pharmaceuticals, a Johnson & Johnson pharmaceutical company.
The purpose of prescription drug Xarelto is threefold: (1) to reduce the risk of blood clots and stroke in patients with atrial fibrillation; (2) to treat pulmonary embolisms and deep vein thrombosis; and (3) and to lessen the risk of blood clots in patients during knee or hip replacement surgery.
Xarelto has been marketed as easier to use than warfarin (coumadin), contending that it is "the first and only once-a-day prescription blood thinner for patients with AFib not caused by heart valve problems, that is proven to reduce the risk of stroke – without routine blood monitoring." Nevertheless, important safety information and warnings were withheld about the dangers of Xarelto.
The dangers associated with Xarelto are not fully known because Xarelto's drug manufacturer has not release this information. Xarelto does not have any antidote. The number of people experiencing side effects from this drug is increasing.
The Institute for Safe Medication Practices (ISMP) did an analysis showing that, in 2013, there were more than 700 adverse event reports submitted to the FDA involved risks associated with Xarelto. In 2013, in Germany, doctors reported 72 deaths from Xarelto use.
Doctors are still prescribing the drug, despite the fact that there may be safer blood-thinners available on the market.
Medication is supposed to be safe. Unfortunately, many drug manufacturers market and distribute their products either without proper testing or warning labels, or both. Many dangerous drug side effects are known to the manufacturers, but they decide to market and sell their drug as safe. The attorneys at Drug Injury Legal Group investigate claims across the country to make sure that drug manufacturers are held accountable for their actions.
In November 2011, the Xarelto warnings were updated to include a boxed warning that "discontinuing Xarelto in patients with non-valvular atrial fibrillation increases the risk of stroke, and spinal/epidural hematoma." A warning was also included that an antidote for Xarelto is not available, placing patients at risk of severe bleeding.
In August 2013, warnings were added to the Xarelo safety label: (1) advising that stopping the use of Xarelto prematurely can increase the risk of thrombotic events, or blood clots; and (2) informing that Xarelto is not recommended for patients with prosthetic heart valves as the safety and efficacy of Xarelto has not been studied in that patient population.
If you or someone you know has experienced a bleeding side effect from the use of Xarelto, including internal bleeding, brain hemorrhaging, blood clots or stroke, please fill out our evaluation form for a free and confidential evaluation of the case, and one of our attorneys will contact you. Or call us toll-free today for a confidential and free case evaluation at 855-571-DRUG.


Landscaping with water is a permaculture exploration led by Ryan Strand. August 20, 9a-11am at the Paonia Community Garden.
This 2016 Permaculture Design class will discuss design and how to apply it your yard or homestead to maximize your space potential and productivity.


Widely regarded as one of Tuscany's smartest and most elegant beach resorts, Forte dei Marmi is the traditional summer retreat for Italian captains of industry, writers and film stars and wealthy visitors from all over Europe. Despite its undeniable appeal to the rich and famous it tries to remain low-key and understated. On the main square and in the back streets there might be luxury boutiques sporting designer goods but there are also family run bakers and butchers, and oligarchs on bicycles patornising them. The beach is Forte die Marmi's crowning glory, a flawless confection of ice-cream coloured beach-huts, perfectly positioned parasols and exquisitely raked white sand.


text:
The space policy of the Barack Obama administration was announced by U.S. President Barack Obama on April 15, 2010, at a major space policy speech at Kennedy Space Center. He committed to increasing NASA funding by $6 billion over five years and completing the design of a new heavy-lift launch vehicle by 2015 and to begin construction thereafter. He also predicted a U.S.-crewed orbital Mars mission by the mid-2030s, preceded by the Asteroid Redirect Mission by 2025. In response to concerns over job losses, Obama promised a $40 million effort to help Space Coast workers affected by the cancellation of the Space Shuttle program and Constellation program.

The Obama administration's space policy was made subsequent to the final report of the Review of United States Human Space Flight Plans Committee, which it had instituted to review the human spaceflight plans of the United States in the post-Space Shuttle era.  The NASA Authorization Act of 2010, passed on October 11, 2010, enacted many of the Obama administration's space policy goals.

Background

In November 2007, the Obama presidential campaign released a policy document delaying NASA's Constellation program by five years to fund education programs. There was concern that any delay would prolong the gap after the Space Shuttle's retirement, when the US would be dependent on the Russian government for access to the International Space Station. Other presidential candidates, including Hillary Clinton, did not support the delay.

In January 2008, the Obama campaign revised its position, supporting the immediate development of the Crew Exploration Vehicle and the Ares I rocket to narrow the gap. However, the new policy was silent on the heavy lift Ares V rocket and missions beyond low Earth orbit.

Obama gave the first major space policy speech of his campaign in Titusville, Florida in August 2008. He subsequently approved a seven-page space plan that committed to target dates for destinations beyond low Earth orbit:

Obama was noted for "having offered more specifics about his plans for NASA than any U.S. presidential candidate in history".

Augustine Commission

The Obama administration instituted the Review of United States Human Space Flight Plans Committee, also known as the Augustine Commission, to review the human spaceflight plans of the United States after the time NASA had planned to retire the Space Shuttle.  Their goal was to ensure the nation is on "a vigorous and sustainable path to achieving its boldest aspirations in space."  The review was announced by the Office of Science and Technology Policy (OSTP) on May 7, 2009.  The report was released on October 22, 2009.

The Committee judged the nine-year-old Constellation program to be so behind schedule, underfunded and over budget that meeting any of its goals would not be possible. The President removed the program from the 2010 NASA budget request and a bi-partisan congress refused to fund it any longer, effectively canceling the program. One component of the program, the Orion crew capsule, was added back into plans, but only as a rescue vehicle to complement the Russian Soyuz in returning Station crews to Earth in the event of an emergency.

In its final report, the Committee proposed three basic options for exploration beyond low Earth orbit, and appeared to favor the third option:

 Mars First, with a Mars landing, perhaps after a brief test of equipment and procedures on the Moon.
 Moon First, with lunar surface exploration focused on developing the capability to explore Mars.
 A Flexible Path to inner Solar System locations, such as lunar orbit, Lagrange points, near-Earth objects and the moons of Mars, followed by exploration of the lunar surface and/or Martian surface.

In his April 15, 2010 space policy speech at Kennedy Space Center announcing the administration's plans for NASA, none of the 3 plans outlined in the Committee's final report was completely selected.  The President rejected immediate plans to return to the Moon on the premise that the current plan had become nonviable.  He instead promised $6 billion in additional funding and called for development of a new heavy lift rocket program to be ready for construction by 2015 with crewed missions to Mars orbit by the mid-2030s.

Space policy speech at Kennedy Space Center
Obama held a space policy speech, titled "Remarks by the President on Space Exploration in the 21st Century", at the Kennedy Space Center on April 15, 2010.  Attendees included then-NASA administrator and Obama appointee Charles Bolden, Apollo 11 Lunar Module Pilot Buzz Aldrin, Representative Suzanne Kosmas (D-FL), Senator Bill Nelson (D-FL), and Representative Sheila Jackson Lee (D-TX), among others.

Modification of Orion
Obama announced the modification of the Orion capsule from its original purpose as a crewed spacecraft for flights to the ISS and the Moon into an emergency escape capsule for the International Space Station, reducing American reliance on the Russian Soyuz spacecraft.

Destinations
The new policy changes from Vision for Space Exploration and Constellation program Moon-first approach to a variety of destinations resembling the flexible path approach:

Obama promoted the idea of a crewed mission to orbit Mars by the mid-2030s with a landing as a follow-up.

Reliance on commercially operated launch vehicles

In a major shift in the function of NASA in American human spaceflight, the Obama administration proposal would rely solely on launch vehicles designed, manufactured, and operated by private aerospace companies, with NASA paying for flights for government astronauts.

Prior to the speech, Obama toured the launch facilities surrounding the SpaceX Falcon 9 launch vehicle with CEO Elon Musk which has received NASA funding through the Commercial Orbital Transportation Services program and could loft NASA astronauts under the new plan. In the speech, Obama referenced the maiden flight of the Falcon 9, which would go on to launch successfully less than one and a half months after the speech:

Extension of ISS operations
Obama also announced an extension of funding for International Space Station operations, 90% complete by mass at the time of the speech but scheduled to be deorbited by as early as 2015 before Obama announced the extension, which will provide funding through 2020.

Heavy-lift launch vehicle
In the speech, Obama announced the development of a new Heavy Lift Vehicle (HLV) to replace the planned Ares V, with the design planned for completion by 2015, two years prior to the Constellation plan, and construction commencing thereafter:

Response

Support

Prior to the speech, SpaceX CEO Elon Musk released a statement praising the proposal and criticizing Senators who had opposed it and former NASA administrator Michael Griffin:

Apollo 11 astronaut Buzz Aldrin released a statement in support of the plan via the White House on February 1, prior to the announcement of the 2011 federal budget, which included the changes Obama announced in the speech:

Criticism

Neil Armstrong, Jim Lovell, and Eugene Cernan, commanders of Apollo 11, Apollo 13, and Apollo 17 respectively, said:

 "When President Obama recently released his budget for NASA, he proposed a slight increase in total funding...the accompanying decision to cancel the Constellation program, its Ares 1 and Ares V rockets, and the Orion spacecraft, is devastating."
 "It appears that we will have wasted our current ten plus billion dollar investment in Constellation and, equally importantly, we will have lost the many years required to recreate the equivalent of what we will have discarded."
 "For The United States, the leading space faring nation for nearly half a century, to be without carriage to low Earth orbit and with no human exploration capability to go beyond Earth orbit for an indeterminate time into the future, destines our nation to become one of second or even third rate stature.  While the President's plan envisages humans traveling away from Earth and perhaps toward Mars at some time in the future, the lack of developed rockets and spacecraft will assure that ability will not be available for many years."
 "Without the skill and experience that actual spacecraft operation provides, the USA is far too likely to be on a long downhill slide to mediocrity."

Robert Zubrin, President of The Mars Society and whose book The Case for Mars first proposed a Shuttle-derived heavy lift vehicle named Ares, lambasted Obama's plans in the New York Daily News:

 "Under the Obama plan, NASA will spend $100 billion on human spaceflight over the next 10 years in order to accomplish nothing"
 "Obama called for sending a crew to a near Earth asteroid by 2025. ... Had Obama not canceled the Ares V, we could have used it to perform an asteroid mission by 2016. But the President, while calling for such a flight, actually is terminating the programs that would make it possible."
 "With current in-space propulsion technology, we can do a round-trip mission to a near-Earth asteroid or a one-way transit to Mars in six months ... Holdren claims that he wants to develop a new electrically powered space thruster to speed up such trips. But without gigantic space nuclear power reactors to provide them with juice, such thrusters are useless, and the administration has no intention of developing such reactors."

Video

Subsequent developments
The Obama administration released its new formal space policy on June 28, 2010. On July 1, 2010, the NASA administrator, Charles Bolden, said in an interview with al-Jazeera: "When I became the Nasa administrator, he [Mr Obama] charged me with three things. One, he wanted me to help reinspire children to want to get into science and math; he wanted me to expand our international relationships; and third, and perhaps foremost, he wanted me to find a way to reach out to the Muslim world and engage much more with dominantly Muslim nations to help them feel good about their historic contribution to science, math, and engineering." This was the first time NASA did not specifically list space exploration as a priority.

The NASA Authorization Act of 2010, passed on October 11, 2010, authorized funds for NASA for fiscal year 2011–2013, and enacted many of his stated space policy goals. A total of $58 billion in funding is called for, spread across three years.

The 2011 budget legislation, passed in April 2011, officially terminated the Constellation program.  The passage of the budget frees NASA to start working on the new initiatives.

In September 2011, details were announced of the Space Launch System, a Shuttle-Derived Launch Vehicle being developed by NASA as a replacement for the Ares I and Ares V rockets of the Constellation program.

The Obama administration cut NASA's planetary-sciences budget by 20 percent in 2013, as part of a restructuring plan, contrary to the recommendations of the National Research Council.

In January 2014 the Obama administration announced it would seek to extend the operational life of the International Space Station by four more years to 2024.

See also
 Shuttle-Derived Heavy Lift Launch Vehicle (2009)
 Space policy of the United States

References

External links

 * Video of speech HD
 Speech transcript at NASA.gov
 Obama's New Space Exploration Plan Includes Major Role for Private Firms – video report by Democracy Now!
 Special Message to the Congress on Urgent National Needs – John F. Kennedy's May 25, 1961 speech to Congress

2010 in American politics
2010 in spaceflight
2010 speeches
Speeches by Barack Obama
Public policy of the Obama administration
NASA oversight
Space policy of the United States
Articles containing video clips
Space policy by United States presidential administration

redpajama_set_name: RedPajamaWikipedia


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