text:
\section{Introduction}
Quadric surface bundles are objects of classical study \cite{staude}.
A special role is played by quadric surface bundles over surfaces
in diverse settings, e.g., \cite{voisin}, \cite{beauville}, \cite{vangeemen},
including questions about (stable) rationality
\cite{ABP}, \cite{HPT}, \cite{schreieder}.

One is interested in the structure of quadric surface bundles,
such as the characterization of smooth models.
Over a surface base, such a result is given in \cite{APS}.

Our main result allows a base of arbitrary dimension.

\begin{thm}
\label{thm.main}
Let $k$ be a perfect field of characteristic different from $2$,
$S$ a smooth projective variety over $k$ of dimension $n$,
and $\pi\colon X\to S$ a morphism of projective varieties which, after base
change $X_{k(S)}\to \Spec(k(S))$ to the generic point of $S$ yields a
smooth quadric surface $X_{k(S)}$.
We assume, over $k$, that resolution of marked ideals on smooth varieties
of dimension $n$ is available.
Then there exists a commutative diagram
\[
\xymatrix{
X' \ar@{-->}[r]^{\varrho_X} \ar[d]_{\pi'} & X \ar[d]^{\pi} \\
S' \ar[r]^{\varrho_S} & S
}
\]
such that
\begin{itemize}
\item $\varrho_S$ is a birational morphism of smooth projective varieties,
\item $\varrho_X$ is a birational map that restricts to an isomorphism over
the generic point of $S$,
\item $\pi'$ is projective and flat and admits a factorization through the
projectivization of a rank $4$ vector bundle on $S'$,
in an \'etale neighborhood
of any point of $S$ taking on one of the normal forms given in Table \ref{table.normalforms}.
\end{itemize}
\end{thm}

\begin{table}
\begin{align*}
K^2-L^2+M^2-N^2&=0,  \\
t_1K^2-L^2+M^2-N^2&=0\ \ \ (n\ge 1),  \\
t_1K^2-t_1L^2+M^2-N^2&=0\ \ \ (n\ge 1),  \\
t_1t_2K^2-t_2L^2+M^2-N^2&=0\ \ \ (n\ge 2),  \\
t_1K^2-L^2+t_2M^2-t_2N^2&=0\ \ \ (n\ge 2),  \\
t_1t_2K^2-t_1L^2+M^2-t_2N^2&=0\ \ \ (n\ge 2),  \\
t_1t_2t_3K^2-t_2L^2+M^2-t_3N^2&=0\ \ \ (n\ge 3),  \\
t_1t_2K^2-t_2t_3L^2+t_3M^2-N^2&=0\ \ \ (n\ge 3).
\end{align*}
\caption{\'Etale local forms of a
quadric surface bundle in Theorem \ref{thm.main}, over affine space
${\mathbb A}^n=\Spec(k[t_1,\dots,t_n])$ over the given field $k$,
with projective coordinates $K$, $L$, $M$, $N$}
\label{table.normalforms}
\end{table}

Another point of view, taken in \cite{AB}, takes as its starting point
the embedding of a quadric surface by its anticanonical linear system
as a degree $8$ del Pezzo surface in ${\mathbb P}^8$, or
\emph{involution surface};
an involution surface is isomorphic to a quadric surface if and only if it
admits a line bundle whose class is half the anticanonical class.
Over a surface base,
involution surface bundles are described in \cite{involution}.

In Section \ref{sec.preliminaries} we recall basic facts about quadric surfaces
and involution surfaces and prove some preliminary results.
In Section \ref{sec.reduction} we give some reduction steps toward the
proof of Theorem \ref{thm.main}.
Section \ref{sec.construction} carries out the construction
of $\widetilde{X}$ and $\widetilde{S}$ in Theorem \ref{thm.main}.
In Section \ref{sec.local} the \'etale local form is justified,
completing the proof of Theorem \ref{thm.main}.
Appendix \ref{app.nonflat} shows that the construction of involution surface
bundles in
\cite{involution} may lead to a non-flat model when attempted
over a base of dimension $>2$.
\subsection*{Acknowledgements}
The author would like to thank his PhD advisor Andrew Kresch for suggesting the study of quadric surface bundles and for the many discussions that followed.
\section{Preliminaries}
\label{sec.preliminaries}
The smooth quadric surface $X_{k(S)}$ uniquely determines a degree
$2$ \'etale $k(S)$-algebra $k(T)$ and an element $\alpha$ of the
Brauer group $\Br(k(T))$, the class of a quaternion algebra that
corestricts to $0$ in $Br(k(S))$ \cite[Example 3.3]{AB}.
Without the condition on corestriction we get a description of a
general involution surface over $k(S)$:
\[ X_{k(S)}\cong R_{k(T)/k(S)}\mathrm{SB}(\alpha), \]
where $\mathrm{SB}(\alpha)$ denotes the Severi-Brauer variety,
the plane conic associated with the
quaternion algebra whose class is $\alpha$,
and $R_{k(T)/k(S)}$ denotes Weil restriction.

The condition on corestriction is an instance of a more general fact
about Weil restrictions of Brauer-Severi varieties.
Geometrically, a Brauer-Severi variety gives rise, upon Weil restriction,
to a variety which is geometrically a product of several copies of
projective space.
On such a variety, a line bundle has a geometric multidegree, a
sequence of integers that is well-defined up to order.

\begin{lemma}
\label{lem.corestriction}
Let $K$ be a field, $L$ a finite \'etale $K$-algebra,
$\alpha\in \Br(L)$, and
$S$ a Brauer-Severi variety over $L$ for $\alpha$.
Set $\beta:=\cores_{L/K}(\alpha)$, and let $d$ be an integer.
Then $d\cores_{L/K}(\alpha)=0$ if and only if
$R_{L/K}(S)$ possesses a line bundle
of geometric multidegree $(d,\dots,d)$.
\end{lemma}

For a Brauer-Severi variety $T\to \Spec(K)$,
the Leray spectral sequence and the fact that $T$, geometrically a
projective space, has geometric Picard group ${\mathbb Z}$ acted upon trivially by
the absolute Galois group of $K$ lead to an exact sequence
\begin{equation}
\label{eqn.lerayT}
0\to \Pic(T)\to {\mathbb Z}\to \ker(\Br(K)\to \Br(T))\to 0
\end{equation}
(see, for instance, \cite[\S 1]{CTO}).
By \cite[V.4.8.3]{giraud},
the element $1\in {\mathbb Z}$ corresponding to the ample generator of the
geometric Picard group of $T$ maps
to minus the class in $\Br(K)$, of which $T$ is a Brauer-Severi variety.

\begin{proof}
To $S$ there is a corresponding central simple algebra over $L$.
A construction in \cite{KO} (see also \cite{ferrand}),
which generalizes the norm of an invertible sheaf \cite[\S 6.5]{EGA2},
supplies the norm algebra, a central simple algebra over $K$
with the Brauer class $\beta$.
We take $T$ to be the associated Brauer-Severi variety.
There is the \emph{twisted Segre embedding}
\[ R_{L/K}(S)\to T, \]
described in \cite{artinbrauerseveri}
in the case $L=K\times\dots\times K$
and straightforward to obtain in the present generality using the functoriality of the
norm construction of \cite{KO}.

Suppose $d\beta=0$.
By \eqref{eqn.lerayT}, there exists a
line bundle on $T$ of geometric degree $d$.
The restriction to $R_{L/K}(S)$ has geometric multidegree $(d,\dots,d)$.

For the converse assertion, we suppose that
$R_{L/K}(S)$ possesses a line bundle of geometric multidegree $(d,\dots,d)$.
Analogous to \eqref{eqn.lerayT} is an exact sequence
\begin{equation}
\label{eqn.lerayR}
0\to \Pic(R_{L/K}(S))\to P\to \ker(\Br(K)\to \Br(R_{L/K}(S)))\to 0;
\end{equation}
where $P\subset {\mathbb Z}^{[L:K]}$ is the group of Galois-invariant multidegrees on
$R_{L/K}(S)$.
By hypothesis, $(d,\dots,d)$ lies in image of the left-hand map of
\eqref{eqn.lerayR}.
We have functorial maps from \eqref{eqn.lerayT} to \eqref{eqn.lerayR},
where $1\in {\mathbb Z}$ is mapped to $(1,\dots,1)\in P$.
This, in turn, is sent by the right-hand map of \eqref{eqn.lerayR} to $-\beta$.
So $d\beta=0$.
\end{proof}

\begin{remark}
\label{rem.arithmeticinvolution}
In \cite[Table 3]{AB} an exhaustive list of
arithmetic possibilities in the case of an involution surface
($[L:K]=2$, $S$ a conic) is given.
\end{remark}

\begin{lemma}
\label{lem.quaterniondescend}
Let $K$ be a field of characteristic different from $2$ with
quadratic extension $L/K$ and class
$\alpha\in \Br(L)$ of a quaternion algebra on $L$.
Then $\cores_{L/K}(\alpha)=0$ if and only if
$\alpha$ is equal to the restriction of the class of a
quaternion algebra on $K$.
\end{lemma}

\begin{proof}
Restriction followed by corestriction is multiplication by $2$.
So,
if $\alpha$ is obtained by restriction from the class of a
quaterion algebra on $K$, then $\cores_{L/K}(\alpha)=0$.
It remains to show, under the assumption that
$\cores_{L/K}(\alpha)=0$, that $\alpha$ is the restriction of the class of a
quaternion algebra on $K$.
We assume $\alpha\ne 0$.

The sequence of cohomology of the Kummer sequence
identifies $H^2(\Spec(K),\mu_2)$ with the
$2$-torsion in the Brauer group $\Br(K)[2]$.
The restriction of scalars $R_{L/K}(\mu_2)$ fits in an exact sequence
\[ 0\to \mu_2\to R_{L/K}(\mu_2)\to \mu_2\to 0. \]
Exactness of the direct image functor under
$\Spec(L)\to \Spec(K)$ \cite[II.3.6]{milne} and the Leray spectral sequence
identify
$H^2(\Spec(K),R_{L/K}(\mu_2))$ with
$H^2(\Spec(L),\mu_2)$, and the long exact sequence of
cohomology identifies the image of the restriction map
\[ \Br(K)[2]\to \Br(L)[2] \]
with the kernel of the corestriction map on $2$-torsion Brauer classes.

Let $\beta\in \Br(K)[2]$, restricting to $\alpha\in \Br(L)$.
Then $\beta$ is a $2$-torsion Brauer class of index $2$ or $4$,
hence is the class of a quaternion division algebra, in which case we are done,
or a biquaterion division algebra, which we suppose now.

Let $d\in K^\times\smallsetminus (K^\times)^2$ be such that
$L\cong K(\sqrt{d})$.
We claim that there exist $a$, $b$, $c\in K^\times$ such that
$\beta=(a,b)+(c,d)$.
The claim implies the lemma, since then
the Brauer class of the quaternion algebra $(a,b)$
restricts to $\alpha$.

According to \cite[Prop.\ 3.2.1]{kahnformesquadratiques},
if an anisotropic quadratic form $q$ over $K$ becomes
isotropic over $K(\sqrt{d})$, then $q$ is similar to
$\langle 1,-d\rangle\perp q'$ for some quadratic form $q'$.
The \emph{Albert form} attached to a pair of quaternion algebras
is anisotropic if and only if the corresponding biquaternion algebra is a
division algebra.
By \cite[Thm.\ 8.1.13]{kahnformesquadratiques},
two Albert forms are similar if and only if they come from the same Brauer class.

We now prove the claim.
Since $\beta$ is the class of a biquaternion division algebra,
restricting to the class of a quaternion algebra in $\Br(L)$,
the Albert form of $\beta$ is anisotropic but becomes isotropic over $L$.
So the Albert form is similar to
$\langle 1,-d,-p,q,r,-dpqr\rangle$ for some $p$, $q$, $r\in K^\times$,
which is similar to the Albert form of
\[ (p,d)\qquad\text{and}\qquad (dpq,dpr). \]
The claim now follows from the result about similarity of Albert forms.
\end{proof}

In Theorem \ref{thm.main},
the existence of a factorization
$\widetilde{X}\to {\mathbb P}({\mathcal E})\to \widetilde{S}$ of $\tilde\pi$
is claimed, with ${\mathcal E}$ a rank $4$ vector bundle on $\widetilde{S}$.
Then, on ${\mathbb P}({\mathcal E})$ we have the line bundles
${\mathcal O}_{{\mathbb P}({\mathcal E})}(n)$, $n\in {\mathbb Z}$, giving rise to line bundles
${\mathcal O}_{\widetilde{X}}(n)$ on $\widetilde{X}$ by restriction.
The relative anticanonical sheaf $\omega_{\widetilde{X}/\widetilde{S}}$ is
isomorphic to the tensor product of
${\mathcal O}_{\widetilde{X}}(-2)$ and
a line bundle pulled back from $\widetilde{S}$.
So our attention is drawn to a line bundle on $\widetilde{X}$
whose square is isomorphic to
$\omega_{\widetilde{X}/\widetilde{S}}^\vee\otimes \tilde\pi^*\mathcal{M}$ for some
line bundle $\mathcal{M}$ on $\widetilde{S}$.

With attention to the extra data of a line bundle on $\widetilde{X}$
we prove the following extension of \cite[Lemma 2.8]{kreschtschinkelbssurf}.
There,
$A$ is a Henselian ring with residue field $\kappa$, and $X$ and $Y$ schemes, smooth and projective over $\Spec(A)$
with finite group
$G$ of order invertible in $\kappa$, acting on
$X$, $Y$, and (compatibly) on $A$,
with trivial action on $\kappa$.
Under the assumption
\begin{equation}
\label{eqn.H1vanishing}
H^1(X_{\kappa}, T_{X_{\kappa}})=0,
\end{equation}
it is shown that every $G$-equivariant isomorphism
$\phi: X_{\kappa} \to Y_{\kappa}$ arises by restriction from a
$G$-equivariant isomorphism $\psi: X \to Y$ over $\Spec(A)$.
We give a version with line bundles as extra data.

\begin{lemma} \label{lem.ext}
With the above notation and, besides hypothesis \eqref{eqn.H1vanishing}, also
\begin{equation}
\label{eqn.moreH1vanishing}
H^1(X_{\kappa}, {\mathcal O}_{X_{\kappa}})=0,
\end{equation}
suppose that we further have line bundles $L_X$ on $X$ and $L_Y$ on $Y$,
together with compatible $G$-actions on them and a $G$-equivariant isomorphism
$L_X|_{X_{\kappa}} \cong \phi^*(L_Y|_{Y_{\kappa}})$.
Then there is also a $G$-equivariant isomorphism $L_X\cong \psi^*L_Y$ of line bundles on $X$.
\end{lemma}

\begin{proof}
By \cite[Corollary 18.6.15]{EGA4}, we may assume that $A$ is Noetherian.
To have a $G$-equivariant isomorphism $L_X\cong \psi^*L_Y$ is equivalent to
having a nowhere vanishing $G$-invariant global section of
$M:=L_X^{\vee} \otimes \psi^*L_Y$.
Since a global section may be averaged with its translates under the
group action, it suffices to show that
\[ H^0(X,M)\to H^0(X_{\kappa},M|_{X_{\kappa}}) \]
is surjective.
Letting $\mathfrak{m}$ denote the maximal ideal of $A$, and
$\widehat{A}$ the completion, we consider the commutative diagram
\[
\xymatrix{
H^0(X,M) \ar[r] \ar[d] & H^0(X_\kappa, M|_{X_\kappa}) \\
H^0(X,M) \otimes_A \widehat{A}  \ar[r]^(0.47)\sim & \underset{n}\varprojlim \,H^0(X, M|_{X_n}) \ar[u]
}
\]
where the bottom isomorphism comes from the Theorem on Formal Functions, and
$X_n$ is obtained from $X$ by base change via $\Spec(A/\mathfrak{m}^n)\to \Spec(A)$.
Now it is enough to show $\underset{n}\varprojlim\, H^0(X, M|_{X_n}) \to H^0(X_\kappa, M|_{X_\kappa})$ is surjective.
We claim that \mbox{$H^0(X, M|_{X_{n+1}}) \to H^0(X, M|_{X_n})$}
is surjective for all $n$.
Fix $n$, and let $d$ denote the dimension of $\mathfrak{m}^n/\mathfrak{m}^{n+1}$
as $\kappa$-vector space.
Then we have a short exact sequence of ${\mathcal O}_X$-modules
\begin{equation*}
0 \to {\mathcal O}_{X_\kappa}^d \to M|_{X_{n+1}} \to M|_{X_n} \to 0,
\end{equation*}
giving a long exact sequence on cohomology:
\begin{equation*}
0 \to H^0(X_\kappa,{\mathcal O}_{X_\kappa})^d \to H^0(X, M|_{X_{n+1}}) \to H^0(X, M|_{X_n}) \to H^1(X_\kappa, {\mathcal O}_{X_\kappa})^d \to \dots
\end{equation*}
From the cohomology vanishing \eqref{eqn.moreH1vanishing} the claim follows.
\end{proof}

\section{Reduction}
\label{sec.reduction}
For a smooth variety $S$, given a Brauer group element $\alpha \in Br(k(S))$, we may consider its \emph{ramification divisor} on $S$, complement of the largest open subscheme of $S$ with a Brauer group element that restricts to $\alpha$. We also use the notion of \emph{root stack} \cite[\S 2]{cadman}, \cite[App. B]{AGV}, specifically the \emph{iterated root stack} along a set of effective Cartier divisors \cite[Def.\ 2.2.4]{cadman}.
\begin{lemma} \label{lem.reduction}
Let $k$ be a perfect field of characteristic different from $2$,
$S$ a smooth projective variety over $k$, $k(T)$ a degree $2$ \'etale $k(S)$-algebra and $T \to S$ the normalization of $S$ in $k(T)$. Let $\alpha \in Br(k(T))$ be a class of a quaternion algebra which corestricts to $0$ in $Br(k(S))$. As in \Cref{thm.main}, we assume that resolution of marked ideals on smooth varieties of dimension equal to $\dim(S)$ is available over $k$. Then there exists a birational morphism $S'\to S$, where $S'$ is smooth projective, such that for $T' \to S'$ the normalization of $S'$ in $k(T)$ we have:
\begin{itemize}
\item the branch divisor $D'=D_1 \cup \dots \cup D_r$ of $T' \to S'$ on $S'$ is smooth.
\item There is a simple normal crossing divisor $D_1 \cup \dots \cup D_r \cup E_1 \cup \dots \cup E_s$ on $S'$, such that for $F_i$'s the pre-images of $E_i$'s on $T'$, $F_1 \cup \dots \cup F_s$ is the ramification divisor of $\alpha$ on $T'$.
\item $\alpha$ is the class of a quaternion algebra over $k(T')$ which is
the restriction of a sheaf of Azumaya algebras of degree $2$ on the iterated root stack $\sqrt{(T',\{F_1,\dots,F_s\})}$.
\item All intersections of triples of divisors $E_1, \dots , E_s$ are empty.
\item The projective representation of
$\mu_2 \times \mu_2$ determined by the sheaf of Azumaya algebras at any component of $F_i \cap F_j$,
$i \neq j$, is faithful.
\end{itemize}
\end{lemma}
Notice that smoothness of the branch divisor on $S'$ implies that $T'$ is smooth.

\begin{proof}
By \Cref{lem.quaterniondescend}, $\alpha$ is the restriction of the class $\alpha_0$ of a quaternion algebra in $\Br(k(S))$. Once a choice of such $\alpha_0$ has been made, we may consider its ramification divisor $E$ on $S$. We also have the branch divisor $D$ of $T \to S$. The proof of \Cref{lem.reduction} is a series of reduction steps on $S$, $D$ and $E$.

As a first reduction step, application of the resolution hypothesis tells us,
\emph{we may suppose that the components of the branch divisor and the
	ramification divisor (for a choice of $\alpha_0$) together form a
	simple normal crossing divisor on $S$}.
We denote by $D_1$, $\dots$, $D_r$ the components of the branch divisor.
The ramification divisor may have components in common with the
branch divisor; we denote its additional components by
$E_1$, $\dots$, $E_s$.
Arguing as in the preparation to the
proof of \cite[Thm.\ 4]{oesinghaus}, we pass to the iterated root stack
along all the divisor components
\[
\mathcal{R}:=\sqrt{(S,\{D_1,\dots,D_r,E_1,\dots,E_s\})}
\]
and obtain a global (cohomological) Brauer class restricting to $\alpha_0$:
\[ \alpha_0\in
\image\big(\Br(\mathcal{R})\to \Br(k(S))\big). \]

Here is the second reduction step:
\emph{we may suppose that there is a simple normal crossing divisor on $S$
	containing the branch divisor, such that the corresponding iterated root stack
	admits a sheaf of Azumaya algebras whose restriction to the generic point is a
	quaternion algebra with Brauer class $\alpha_0$}.
This uses Raynaud-Gruson flatification \cite{raynaudgruson}, as in
the proof of \cite[Thm.\ 1.2]{kreschtschinkelbssurf}.

The quaternion algebra over $k(S)$ spreads out to a sheaf
of Azumaya algebras on some nonempty open $U\subset S$,
disjoint from the ramification divisor and thus identified with
an open substack of $\mathcal{R}$.
Better, we associate to the (unique by \cite[Prop.\ 2.5(iv)]{antieaumeier})
Brauer class on $\mathcal{R}$ restricting to $\alpha_0$ a gerbe
\[ \mathfrak{G}\to \mathcal{R} \]
banded by ${\mathbb G}_m$, and the sheaf of Azumaya algebras determines by the
machinery of nonabelian cohomology, canonically, a rank $2$ vector bundle on
$\mathfrak{G}\times_{\mathcal{R}}U$ whose projectivization descends to yield
the associated Brauer-Severi scheme over $U$ (see the proof of
\cite[Thm.\ IV.2.5]{milne}).
By \cite[Cor.\ 15.5]{LMB},
this vector bundle is the restriction of a coherent sheaf
$\mathcal{E}$ on $\mathfrak{G}$.
To $\mathcal{E}$ there is a Fitting ideal sheaf, which descends to a
coherent sheaf of ideals on $\mathcal{R}$, corresponding to a closed substack
disjoint from $U$, such that upon blowing up to obtain
$\mathcal{R}'\to \mathcal{R}$
there is a vector bundle on $\mathfrak{G}\times_{\mathcal{R}}\mathcal{R}'$
which restricts to the given vector bundle
on $\mathfrak{G}\times_{\mathcal{R}}U$.

We wish to apply Raynaud-Gruson flatification to $\mathcal{R}'\to S$.
While Raynaud-Gruson flatification is defined for schemes, not stacks,
$\mathcal{R}$ is Zariski locally over $S$
a stack quotient by some power of $\mu_2$,
hence $\mathcal{R}'$ enjoys similar local quotient presentations,
and the machinery of Raynaud-Gruson flatification is local in nature
(see, specifically, \cite[(5.3.3)]{raynaudgruson}), so that we may obtain
a coherent sheaf $\mathcal{I}$ of ideals on $S$, corresponding to a
closed subscheme disjoint from $U$, whose blow-up
$\Proj(\bigoplus \mathcal{I}^j)$
has the flatification property: the stack-theoretic closure of $U$ in
$\mathcal{R}'\times_S\Proj(\bigoplus \mathcal{I}^j)$ is
flat over $\Proj(\bigoplus \mathcal{I}^j)$.
The same holds with $\Proj(\bigoplus \mathcal{I}^j)$ replaced by any
projective variety $\widetilde{S}$ with birational morphism to $S$, restricting to an
isomorphism over $U$ and factoring through $\Proj(\bigoplus \mathcal{I}^j)$.
We may apply resolution to the ideal
$\mathcal{I}$ with marking
$D_1$, $\dots$, $D_r$, $E_1$, $\dots$, $E_s$,
to obtain such $\widetilde{S}$ via a sequence of blow-ups
\[ \widetilde{S}\to \dots \to S, \]
each along a smooth center that has normal crossing with the marking
(to which, after each blow-up, the exceptional divisor gets added).

We consider the branch divisor on $\widetilde{S}$
(of its normalization in $k(T)$), whose components
are the proper transforms of
$D_1$, $\dots$, $D_r$ together with
some subset of the components of the exceptional divisors.
There is also the ramification divisor of
$\alpha_0$ on $\widetilde{S}$.
The components of these, together, form a simple normal crossing divisor
on $\widetilde{S}$.
The normalization morphism of $\mathcal{R}\times_S\widetilde{S}$ factors through
the stack-theoretic closure of $U$ in
$\mathcal{R}'\times_S\widetilde{S}$, because the morphism from the stack-theoretic closure
to $\mathcal{R}\times_S\widetilde{S}$ is representable and finite birational.
The iterated root stack of $\widetilde{S}$ along the components of its
branch and ramification divisors factors through $\mathcal{R}\times_S\widetilde{S}$
hence as well through the normalization,
and we have achieved the second reduction step.

We give a third reduction step: \emph{we may suppose that the
	branch divisor on $S$ is smooth, the
	ramification divisor on $T$ of the Brauer class $\alpha$ is the
	pre-image of a divisor on $S$ which has no common components with
	the branch divisor, the two together forming a simple normal crossing divisor
	on $S$, and $\alpha$ is the class of a quaternion algebra over $k(T)$ which is
	the restriction of a sheaf of Azumaya algebras on the iterated root stack of $T$
	along the components of the ramification divisor}.
Notice that smoothness of the branch divisor on $S$ implies that
$T$ is smooth.

Starting from the outcome of the second reduction step,
with branch divisor $D_1\cup\dots\cup D_r$ and simple normal crossing divisor
$D_1\cup\dots\cup D_r\cup E_1\cup\dots\cup E_s$ on $S$,
it is easy to achieve
smoothness of the branch divisor by repeatedly blowing up a
nonempty locus of the form $D_i\cap D_j$ with $i\ne j$.
By applying Steps 2 and 3 of the proof of \cite[Thm.\ 3.1]{oesinghaus} to
components of intersections of pairs, respectively triples of the
divisors $E_1$, $\dots$, $E_s$, we may suppose that the projective
representation of $\mu_2\times \mu_2$
determined by the sheaf of Azumaya algebras at any component of
$E_i\cap E_j$, $i\ne j$, is faithful, and all intersections of
triples of divisors $E_1$, $\dots$, $E_s$ are empty.
Let $F_i$ denote the pre-image of $E_i$ under $T\to S$.
So, we have a finite \'etale morphism
\[
\sqrt{(T,\{F_1,\dots,F_s\})}\to \sqrt{(S,\{D_1,\dots,D_r,E_1,\dots,E_s\})},
\]
by which the sheaf of Azumaya algebras pulls back to yield, on
$\sqrt{(T,\{F_1,\dots,F_s\})}$, a sheaf of Azumaya algebras
whose restriction to $k(T)$ has Brauer class $\alpha$.
Application of Step 4 of loc.\ cit.\ completes the third reduction step.

Composing the three reduction steps into a single birational morphism $S' \to S$ we achieve the desired properties.
\end{proof}
\section{Construction} \label{sec.construction}
With the starting data of \Cref{thm.main}, we now apply \Cref{lem.reduction} in the case when $k(T)$ is the \emph{discriminant extension} parametrizing rulings of the quadric surface $X_{k(S)}$ and $\alpha \in Br(k(T))$ is the corresponding class of a quaternion algebra that corestricts to zero \cite[Example 3.3]{AB}.

The sheaf of Azumaya algebras on $\sqrt{(T',\{F_1,\dots,F_s\})}$ of \Cref{lem.reduction} determines a smooth ${\mathbb P} ^1$-fibration $\Omega\to \sqrt{(T',\{F_1,\dots,F_s\})}$. Together with the finite \'etale morphism $\sqrt{(T',\{F_1,\dots,F_s\})}\to \sqrt{(S',\{D_1,\dots,D_r,E_1,\dots,E_s\})}$, we get, by restriction of scalars, a smooth $({\mathbb P} ^1 \times {\mathbb P} ^1)$-fibration $\Phi\to \sqrt{(S',\{D_1,\dots,D_r,E_1,\dots,E_s\})}$. We see that $\Phi_{k(S)}$ is isomorphic to $X_{k(S)}$, hence we have a birational map $\Phi \dashrightarrow X$ restricting to an isomorphism over the generic point of $S$.  Over the generic point of $S'$, hence as well over a dense open subscheme of $S'$, we have a smooth fibration in quadric surfaces with relatively very ample line bundle of degree 2. This line bundle is the restriction of a line bundle $L$ on the whole smooth $({\mathbb P} ^1 \times {\mathbb P} ^1)$-fibration by \Cref{lem.invertiblesheafextend} below.

\begin{lemma}
\label{lem.invertiblesheafextend}
Let $X$ be a regular Noetherian algebraic stack, and let
$U\subset X$ be an open substack.
Then the restriction map $\Pic(X)\to \Pic(U)$ is surjective.
\end{lemma}

\begin{proof}
There is no loss of generality in supposing $U$ to be dense in $X$.
By \cite[Cor.\ 15.5]{LMB},
any coherent sheaf on $U$ is isomorphic to the restriction
of a coherent sheaf on $X$.
Let an invertible sheaf on $U$ be given,
isomorphic to the restriction of coherent sheaf $\mathcal{F}$ on $X$.
We may replace $\mathcal{F}$ by its double dual without changing the
isomorphism type of the restriction to $U$, so by
\cite[Cor.\ 1.2]{hartshorne} there is no loss of generality in
supposing $\mathcal{F}$ to be reflexive.
Then $\mathcal{F}$ is invertible by \cite[Prop.\ 1.9]{hartshorne}.
\end{proof}

Recall that $D_1 \cup \dots \cup D_r \cup E_1 \cup \dots \cup E_s$ is a simple normal crossing divisor on $S'$, where $D_1 \cup \dots \cup D_r$ is the branch divisor of the normalization $T' \to S'$ on $S'$. We now call $E_1 \cup \dots \cup E_s$ the \emph{ramification divisor} on $S'$ (note that this is not necessarily the ramification divisor of a class of a quaternion algebra in $\Br (k(S'))$ that restricts to $\alpha$ as we saw in Lemma \ref{lem.reduction}). In Lemma \ref{lem.reduction}, we achieved that the branch divisor is smooth and all intersections of triples of components of the ramification divisor are empty. Now, by \cite[Lemma 2.8]{kreschtschinkelbssurf}, the \'etale local isomorphism type of $\Phi$ is uniquely determined
by the action of the stabilizer group, which is $\mu_2$ generically along
the branch and ramification divisors,
$\mu_2\times \mu_2$ generically along an intersection
(of the branch and ramification divisors, or of two components
of the ramification divisor), and is
$\mu_2\times \mu_2\times \mu_2$ at points of the branch divisor
that lie on two components of the ramification divisor.
Twisting $L$ by components of the gerbe over the branch divisor (i.e., the reduced pre-image
of $D_1 \cup \dots \cup D_r$) as appropriate, we may suppose that the space of
invariant sections of $L$ on fibers of $\Phi$ away from the ramification divisor
always has dimension $> 1$.
We may furthermore suppose that the space of
invariant sections of $L$ is nontrivial on fibers of $\Phi$ over the ramification divisor:
this potentially fails on components with $\mu_2 \times \mu_2 \times \mu_2$ stabilizer, which we
blow up, then after blowing up the intersection of the new exceptional divisor with
the proper transform of the gerbe over the branch divisor and twisting $L$ by both new
exceptional divisors there are nontrivial invariant sections of $L$ on all fibers of $\Phi$.

Let $S'^\circ$ be the complement of $S'$ by all the intersections of divisors above, which comprise a locus that, if nonempty, has codimension $2$. We use the same names $D_i$ and $E_j$ (and branch and ramification divisor, respectively) for their intersections with $S'^\circ$ and let $H$ be their union viewed as a divisor on $S'^\circ$. As $H$ is regular, the iterated root stack $\sqrt{(S'^\circ,\{D_1,\dots,D_r,E_1,\dots,E_s\})}$ is just $\sqrt{(S'^\circ, H)}$, for which we have a simple local description by \cite[Section 2.4]{kreschtschinkelbssurf}: $\sqrt{(S'^\circ, H)}$ is, Zariski-locally on $S'^\circ$, the quotient stack of $\Spec({\mathcal O}_{S'^\circ}[t]/(t^2-f))$ by the scalar action of $\mu_2$ on $t$, where $f$ is the regular function locally defining $H$. Let ${\mathcal G}$ be the gerbe of the root stack $\sqrt{(S'^\circ, H)}$, $\Phi^\circ \to \sqrt{(S'^\circ,H)}$ the pullback of the ${\mathbb P}^1\times {\mathbb P}^1$-fibration $\Phi$, and let ${\mathcal F}$ be the fiber of $\Phi^\circ$ over the gerbe. This is summarized in the diagram below, where each square is cartesian apart from the bottom middle one. Then the fibers over points $x \in H$ with residue field $\kappa(x)$ are copies of $B\mu_{2,\kappa(x)}$ in ${\mathcal G}$ and copies of $[{\mathbb P}^1\times {\mathbb P}^1/ \mu_{2, \kappa(x)}]$ in ${\mathcal F}$, where the action of $\mu_2$ depends on whether $x$ is a point of the branch divisor or the ramification divisor.

\[
\xymatrix{
[{\mathbb P}^1\times{\mathbb P}^1/\mu_{2, \kappa(x)}] \ar[r] \ar[d] & {\mathcal F} \ar[r] \ar[d] & \Phi^\circ \ar[r] \ar[d] & \Phi \ar[d] \\
B\mu_{2,\kappa(x)} \ar[r] \ar[d] & {\mathcal G} \ar[r] \ar[d] & \sqrt{(S'^\circ, H)} \ar[r] \ar[d] & \sqrt{(S',\{D_1,\dots,D_r,E_1,\dots,E_s\})} \ar[d] \\
\Spec \kappa(x) \ar[r] & H \ar[r] & S'^\circ \ar[r] & S'
}
\]

As in the proof of \cite[Prop.\ 3.1]{kreschtschinkelbssurf}, we will perform a construction by blow-up, contraction and descent, yielding quadric surface bundle $B^\circ$ over $S'^\circ$. We then canonically extend this over the codimension $2$ locus to the whole of $S'$ \cite[\S 9]{kreschtschinkelbssurf}.

Over the points of the branch divisor, $\mu_2$ acts by swapping the factors of ${\mathbb P}^1\times{\mathbb P}^1$. We blow up the fixed points under this action, the diagonals $\Delta$ of ${\mathbb P}^1\times {\mathbb P}^1$.

Over the points of the ramification divisor, $\mu_2$ acts by $((s_0:s_1),(t_0:t_1)) \mapsto ((s_0:-s_1),(t_0:-t_1))$ for the coordinates on ${\mathbb P}^1\times{\mathbb P}^1$. We have four fixed points. The restriction of $L$ to a fiber is ${\mathcal O}(1,1)$, with $\mu_2$ action given by either multiplication by $-1$ on $s_0\otimes t_1$ and $s_1\otimes t_0$, multiplication by $1$ on $s_0\otimes t_0$ and $s_1 \otimes t_1$, or the opposite, with $1$ and $-1$ swapped. Global sections on the quotient stack are the invariant sections, which form a pencil of conics on ${\mathbb P}^1\times{\mathbb P}^1$. The base locus is $\{((0:1),(1:0)), ((1:0),(0:1))\}$ for the first action and $\{((0:1),(0:1)), ((1:0),(1:0))\}$ for the second (together they are the four fixed points of the action on ${\mathbb P}^1\times{\mathbb P}^1$). Now we blow up the movable fixed points.

Let $\widetilde{\Phi}$ be the combined blowup, $E$ the exceptional divisor, and $F$ the proper transform of ${\mathcal F}$. On $F\subset \widetilde{\Phi}$ the action of $\mu_2$ is nontrivial. So we need to contract $F$; for this we follow \cite[Proposition A.9]{kreschtschinkelbssurf}.
Let $L_0$ be some appropriate power $a$ of the pullback of $L$ to $\widetilde{\Phi}$ twisted by $F$, so that $L_0$ is ample. We can verify locally over suitable \'etale charts of the root stack that this line bundle gives us the desired contraction.

If $x$ lies on the branch divisor, we have the fiber $F_x\cong {\mathbb P}^1\times{\mathbb P}^1$. The normal bundle $N_{F_x/\widetilde{\Phi}_x}$ is
\begin{equation*}
{\mathcal O}_{\widetilde{\Phi}_x}(F_x)|_{F_x}\cong {\mathcal O}_{\widetilde{\Phi}_x}(-E_x)|_{F_x} \cong {\mathcal O}_{F_x}(-\Delta)\cong {\mathcal O}_{F_x}(-1,-1).
\end{equation*}
Now $L_0|_{F_x}$ is ample and has bidegree $(a-1,a-1)$ so by \cite[Example A.2 (ii)]{kreschtschinkelbssurf}, fiberwise $F_x$ is contracted to a point.

Over points of the ramification divisor, $F$ is a a degree 6 del Pezzo surface, which contracts to a line as in \cite[Step (i) in Section 2.5]{involution}.

Now we have semiample $L_0((a-1)F)=(L(F))^a$ on $\widetilde{\Phi}$, hence semiample $L(F)$, contracting $F$ to some locus in $B^\circ$, where $B^\circ$ has trivial action of $\mu_2$. The contraction yields, as well, a line bundle $M^\circ$ on $B^\circ$ that pulls back along the contraction to $L(F)$.

As $B^\circ$ has trivial action of $\mu_2$ on it, it descends by \cite[Proposition 2.5]{kreschtschinkelbssurf} to a quadric surface bundle $f^\circ:X'^\circ \to S'^\circ$, together with a line bundle $N^\circ$ on it that pulls back to $M^\circ$.

Finally, we can extend $X'^\circ$ across the codimension $2$ locus to the whole of $S'$ as in the final step of \cite[\S 9]{kreschtschinkelbssurf}. Letting $\iota\colon S'^\circ \to S'$ be the inclusion, we let $N=(\iota \circ f^\circ)_*N^\circ$ and consider the Zariski closure of $X'^\circ$ in ${\mathbb P}(N^\vee)$, call it $X'$. We get a quadric surface bundle $X'\to S'$. This is the map $\pi'$ of \Cref{thm.main}.

\section{\'Etale local normal forms} \label{sec.local}
We now want to exhibit models for $\Phi \to \sqrt{(S'^\circ,\{D_1,\dots,D_r,E_1,\dots,E_s\})}$, \'etale local over $S'$, on which the construction of \Cref{sec.construction} can be carried out explicitly in coordinates. \'Etale local form of $\Phi$ at points $x\in S'$ depends on $x$.

In case of $x\in S'$, away from
the branch and ramification divisors, the \'etale local form is
given by the first entry in Table \ref{table.normalforms}.

Suppose $x$ lies on the branch divisor, but not on the ramification divisor.
Then we consider ${\mathbb A}^n=\Spec(k[t_1,\dots,t_n])$ with cover,
again ${\mathbb A}^n$, gotten by
adjoining $s_1$ with $s_1^2=t_1$.
Let $Q$ denote the quadric defined by
\[ A^2-B^2+C^2-D^2=0 \]
in ${\mathbb P}^3=\Proj(k[A,B,C,D])$.
We let $\mu_2$ act on the coordinate $s_1$ of the cover ${\mathbb A}^n$, and
on the coordinate $A$ of $Q$.
The construction, applied to
\[
\xymatrix{
[({\mathbb A}^n\times Q)/\mu_2] \ar[d] \\
[{\mathbb A}^n/\mu_2] \ar[d] \\
{\mathbb A}^n
}
\]
proceeds by blowing up the locus defined by $s_1=A=0$ and contracting the
proper transform of $Q$ over $s_1=0$ to obtain the map
\[ (s_1,t_2,\dots,t_n,A:B:C:D)\mapsto (t_1,\dots,t_n,A:s_1B:s_1C:s_1D). \]
This descends to give the quadric surface bundle over ${\mathbb A}^n$,
whose defining equation appears as the
second entry in Table \ref{table.normalforms}.

Other cases are handled similarly, with an appropriate number of
$\mu_2$ factors acting on the same number of coordintes $s_i$, where
$s_i^2=t_i$, and a suitable action of the product of $\mu_2$ factors on
the coordinates $A$, $B$, $C$, $D$.
We list the maps that arise out of the construction in each case:
\begin{align*}
(s_1,t_2,\dots,t_n,A:B:C:D)&\mapsto (t_1,\dots,t_n,A:B:s_1C:s_1D), \\
(s_1,s_2,t_3,\dots,t_n,A:B:C:D)&\mapsto
(t_1,\dots,t_n,A:s_1B:s_1s_2C:s_1s_2D), \\
(s_1,s_2,t_3,\dots,t_n,A:B:C:D)&\mapsto
(t_1,\dots,t_n,s_2A:s_1s_2B:s_1C:s_1D), \\
(s_1,s_2,t_3,\dots,t_n,A:B:C:D)&\mapsto
(t_1,\dots,t_n,A:s_2B:s_1s_2C:s_1D), \\
(s_1,s_2,s_3,t_4,\dots,t_n,A:B:C:D)&\mapsto
(t_1,\dots,t_n,A:s_1s_3B:s_1s_2s_3C:s_1s_2D), \\
(s_1,s_2,s_3,t_4,\dots,t_n,A:B:C:D)&\mapsto
(t_1,\dots,t_n,s_3A:s_1B:s_1s_2C:s_1s_2s_3D).
\end{align*}
The respective images correspond to
the remaining entries in Table \ref{table.normalforms}.

redpajama_set_name: RedPajamaArXiv


text:
Banksy Becomes Artist in Residence for New York City
This is from October 2013, but it's still pretty cool.
Uploaded by banksyfilm on 2014-06-11.
tags: banksy, graffiti, art, street art, new york, new york city, NY, NYC, artist, graffiti artisit, bristol, british, english, artwork
The Most Amazing Example of Multi-tasking . . . Ever.
Although this is my photography site, this post has absolutely nothing to do with photography. You may or may not know that I'm actually a professional musician by trade. I play drums and this guy has blown me away with his ability to multi-task. I play a lot of different styles of music, including afro-cuban stuff which definitely pushes the hand-foot coordination, but what he's doing is something different, not just putting things into place rhythmically, but also melodically while at the same time singing lyrics and not just a melody. It's impressive, very impressive. I'm definitely a fan of Paris Monster, this New York based drum and bass duo is pretty amazing. To see what I'm talking about check out the live performance of "Vision Complete."
Paris Monster - Josh Dion (drums/vocals/synths) & Geoff Kraly (bass)
tags: paris monster, band, music, musician, musicians, drums, bass, duo, synth, vocals, josh dion, geoff kraly, video, youtube, live, vision complete, new york, nyc, new york city, performance, gig, awesome, amazing, multi-task, multitask, drumming, coordination, stumbleupon, reddit, impressive
categories: Music, Video
I just saw this video on the Vimeo Staff Pick page and it made me miss the hustle and bustle of New York.
District7Media.net
@TheVoder
@District7M
Midtown New York City is the place to be. The lights, cars, traffic, architecture and people make the city come to life as a living, breathing, organism that never sleeps. Take a trip with this time-lapse production into the city and see the sights and sounds that create this magnificent destination.
For us, this time-lapse production was produced using more than 50,000 still frames, shot over the course of 6 months traveling back and forth from Washington DC to New York City while we shot for clients. Our main goal was to capture the central part of the city in both visuals and audio. The sound effects used in this production were taken from the various locations we shot and were added in to give you the feeling of actually being there.
There were multiple times during this shoot that we were chased off, either by cops or the cold. The subway shots were particularly difficult to get, especially in the wake of the Boston bombings. We were led out and in some cases followed by police officers or MTA officials who seemed intent on getting us for using tripods.
The music is performed by the talented Danny McCarthy (Fracture)
(Music has been licensed through Killertracks.com)
***Licensing note***
The majority of this footage is own exclusively by 3rd party clients and we can not license out certain shots. If you're interested in licensing non-exclusive material please contact Andrew at ageraci@district7media.net or call us at 513-259-9231
Follow us here;
facebook.com/District7Media
Our gear consisted of the following;
(4) Canon 5D Mark III's
(2) Canon 16-35mm f/2.8 II
Canon 24-105mm f/4
Canon 70-200mm f/2.8 II
(2) Dynamic Perception Stage Zero's with upgrade stepping motors dynamicperception.com
Emotimo TB3 - Orange and Black emotimo.com/
This production would not be possible without the help, assistance, guidance and inspiration of some of the most talented people out there;
Arthur Breese, Jimmy Shea, Gregg Lehrman, Jay Burlage, CA Time-lapse Crew (Joe Capra, Colin Rich, Andrew Walker, Matt Givot, Chris Pritchard, Michael Shainblum, Eric Hinds, Ben Canales, Sheldon Neil) Richard Bently, Patryk Kizny and the rest of the time-lapse community.
MIDTOWN - Cinematography, production and motion graphics by Andrew "Drew" Geraci
©2013 District 7 Media All Rights Reserved. No images, video or parts of this production may be used without authorization. This production is for portfolio use only.
Source: https://vimeo.com/thevoder
tags: vimeo, video, vimeo staff pick, time lapse, timelapse, time-lapse, midtown, nyc, ny, new york, manhattan, thevoder, the voder, voder, district7, district7m, new york city, city, canon, camera, canon 5D, 5D, dynamic perception, emotimo, andrew geraci, drew geraci, district 7 media, production, danny mccarthy

redpajama_set_name: RedPajamaCommonCrawl


text:
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text:
Uber scores record $3.5 billion from Saudi Arabia
By Marisa Kendall | mkendall@bayareanewsgroup.com | Bay Area News Group
PUBLISHED: June 1, 2016 at 7:19 pm | UPDATED: August 17, 2016 at 4:50 am
SAN FRANCISCO — Uber said Wednesday it has raked in $3.5 billion from Saudi Arabia's national investment arm, a deal that sets a record as the largest U.S. funding round for a private, venture-backed company.
The massive investment is remarkable even for Uber, which consistently has raised staggering amounts of cash at sky-high valuations while undergoing explosive global growth. Under terms of the deal, Uber will maintain its valuation of $62.5 billion and its position as the most valuable private company in the world.
"We appreciate the vote of confidence in our business as we continue to expand our global presence," Uber co-founder and CEO Travis Kalanick wrote in an emailed statement. "Our experience in Saudi Arabia is a great example of how Uber can benefit riders, drivers and cities and we look forward to partnering to support their economic and social reforms."
The deal shatters the previous record for a private U.S. funding round, according to Ben Veghte, vice president of communications for the National Venture Capital Association. Last month Snapchat raised $1.8 billion, and San Francisco-based Uber has raised multiple previous rounds topping $1 billion.
The ride-booking platform's most recent funding round is even more noteworthy given that many other tech companies are struggling to raise cash as investors tighten their belts, wary of inflated "unicorn" valuations. Companies including Foursquare and Jawbone have had to accept slashed valuations in order to raise more capital.
The Saudi Arabian Public Investment Fund expects the $3.5 billion investment in Uber to aid the country in its goals of moving away from a dependence on oil, expanding opportunities for women and encouraging entrepreneurship. The country announced an ambitious plan in April dubbed Saudi Arabia's Vision 2030, which aims to lower the unemployment rate, diversify the economy and increase women's participation in the workforce from 22 to 30 percent. Women cannot drive in Saudi Arabia, and Uber says 80 percent of its riders there are women.
"The Saudi Arabian Public Investment Fund is excited to have completed this important strategic investment with Uber, an extraordinary company with an inspiring mission," the fund's managing director, Yasir Al Rumayyan, wrote in a statement. "We've seen firsthand how this company has improved urban mobility around the world and we're looking forward to being part of that progress."
The deal gives Al Rumayyan a seat on Uber's board. Saudi Princess Reema Bander al-Saud, a women's advocate and entrepreneur, recently joined Uber's public policy board.
Brendan Connaughton of ClearPath Capital Partners said Uber may also be using the deal as a way to gain leverage overseas.
"I think part of this is optics and showing that they're a global company," he said. "They've had their challenges internationally."
As Uber strives for global dominance, it has struggled in China, where competitor Didi Chuxing holds a large market share.
From Saudi Arabia's perspective, Connaughton said the investment most likely was intended to "make a splash" and grab the world's attention. That probably factored into the decision to pour all $3.5 billion into one company, instead of dividing it among several investments, he said.
As for Uber, it doesn't appear that the company needed the money, Connaughton said. If Uber did need cash, he said, this deal would have come at a lower valuation. The company says the investment brings its total cash and credit line to more than $11 billion.
"They're not hurting for money," Connaughton said. "They've got capital. Don't think it was a capital call."
Marisa Kendall covers startups and venture capital. Contact her at 408-920-5009 and follow her at Twitter.com/marisakendall.
Marisa Kendall
Marisa Kendall covers housing for the Bay Area News Group, focusing on the impact local companies have on housing availability in the region. She's also written about technology startups and venture capital for BANG, and covered courts for The Recorder in San Francisco. She started her career as a crime reporter for The News-Press in Southwest Florida.
Follow Marisa Kendall	@MarisaKendall


The Trevor Lawson story is testament to the fact that you don't need a background in racing or a lifetime love of the sport to carve out a career in wagering. It should also serve as inspiration to anyone out there thinking of pursuing the racing dream that it's never too late to try.
Racing wasn't a part of Lawson's life growing up. Golf was his passion from about the age of 14 and he showed great natural talent so followed the golfing dream for a few years. He was good enough to win several pro-ams, but never quite good enough to take the next step.
His first job after leaving school was as a trainee pro at Long Island Golf Club in Frankston. This is where he was first exposed to the world of gambling. Like most golf clubs at the time, there was an SP bookie behind the bar, and he would do a roaring trade on Saturdays.
This was a losing proposition for Lawson, who like most recreational punters, was just doing it as a hobby and for the enjoyment. Luckily for Lawson, there was also plenty of betting done on the golf course. He would happily play members or guests for his weekly wage or more per game and this would prove a handy source of income.
He would go along with his boss to the dogs or trots at night, bet on the horses in the day and gamble on golf. He had well and truly been bitten by the bug. Towards the end of his 16 years as a golf pro, he began to ponder what was next after golf without many other qualifications.
His trips to race meetings became more frequent and he started to become friendly with a very smart punter that was involved with one of the leading punting syndicates at the time. He would become somewhat of a mentor to Lawson over the next few years.
Through meeting him, Lawson's 'eyes would be opened' to a new way of doing things. The syndicate was one of the most successful in Australia and operated by using one of very few computer databases dedicated to racing at the time.
It was a similar style system to what Don Scott had written about and Lawson would soon read about. Each horse would be given a weight rating for each run based off numerous factors.
This appealed to Lawson, as he'd always had an analytic, structured way of approaching things, and he could see how lucrative it could be through the success of this syndicate.
He would start to compile his own ratings, but his database was a collection of years' worth of Sportsmans and racebooks that would have his notes and figures scribbled down in them.
He started working with another professional punter and would learn as much as he could from him and others. He would take what he learned and refine how he would come up with his own ratings and after two years felt confident enough to go out and give it a try on his own.
He purchased the Southcoast database and would put all his notes, video comments and ratings into there. The final rating for a horse's run was in kilograms, which could also be represented as lengths.
Lawson has a thorough process when going about giving a numerical value to each runner. It starts with a number that is generated using the horse's beaten margin. A figure for the weight is given based off a scaled compression of the weights carried. A figure for the class of the race (that is being constantly refined) is then adjusted for the early pace in the race.
The horse's position in the run is then given a bonus or penalty depending on how suited it was to this early pace. The video comments also get broken down to a numerical value in kilograms.
All of these steps and many other factors go into giving a horse a rating. Quantifying these factors, and the different ways you could go about it and then weighting them accordingly is the true skill in rating a horse and one that Lawson has spent the past 15 years honing.
When it is time to do the form for a race, the base number that is selected for each runner is then adjusted for numerous more factors. It is done by giving a bonus or penalty (in kilograms) for every variable that Lawson considers important in a race.
The map and what sort of run each horse will get is the first thing that gets taken into account. Along with this a rating for the jockey is given significant weighting. A top jockey such as a Bowman or Williams could get a bonus of approximately 1.5kg and the lesser jockeys could get a penalty of up to 3-4 kg, which gives an indication of just how important the jockey riding is to Lawson.
The other major influence on his rating comes from something that Lawson considers one of his main edges over the marketplace in his years in wagering. He is a huge believer in track bias and the affect it has on the outcome of races, and believes it is underplayed in the market.
This was one of the key lessons learnt from his mentor, who was among the first to try to mathematically account for bias in racing. Lawson, or someone he trusts, will walk every track the day before any meeting he bets on. Not many people do it, and it is a real skill in itself for those that do actually take the time to do it.
Sometimes, it's just for 'peace of mind' he says, as historical records can give a good indication of bias, but 'it gives you confidence when you can confirm it by walking the track prior'.
Other times, however, by walking regularly, he can be the first to pick up on any potential change in pattern that may occur.
Lawson isn't a believer in adjusting during the day. If he gets the call wrong on the track and how it will play, he would rather 'stop and say I was wrong and be done for the day'.
"There are plenty of other days to bet on. It's just one meeting amongst many," he said.
Other factors that get a bonus or penalty include a possible improvement early in a preparation or being suited – or not – to going up or back in distance. Lawson is also a big believer in horses and trainers tending to follow patterns in their preparations. If a horse has improved significantly third-up at its past two campaigns, then Lawson will factor in a likely improvement to its next third-up run.
Once he is happy with the weight rating he has given each horse, Lawson then goes about turning this rating into a price to 90 per cent. He has different algorithms that compress the rating at differing ratios, depending on the type of race and size of the field.
"A horse that is rated three lengths behind the top horse in a weight-for-age race with a small field, should be priced much longer than a horse with a similar rating in a bigger field with more variables," Lawson said.
Even though manual adjustments are made to the rating along each step of the process, the fact that the final price is generated automatically is something Lawson considers to be an edge when it comes to his wagering.
He said: "It stops you being clouded with horses. Once I have given it the rating and all the bonuses I can and the price comes out at $3, I just can't take $2.50.
After he has his prices it's time to bet. The type of the race means little (other than avoiding 2YO races with limited public information). It's all about the size of the overlay on his ratings compared to what's available.
He generally backs anywhere up to seven runners in a race and is happy to take down to $2.50 for his bracket of runners per race. Some runners will be bigger winners than others and some will be just to break square on the race. He finds that backing multiple runners in a race like this allows him to reduce the variance.
The art of taking the best possible price has become much harder in the current landscape. Previously the betting ring at the track was the one and only marketplace. It was easy to know the familiar faces and know when a betting move was occurring and secure top odds.
Now with the marketplace being online, so many moves happen anonymously and it's hard to tell the right bets from the wrong bets.
"Nowadays a horse can be $17 into $16 with five minutes to go and you take the $17 and by jump time it's 30s. That was unheard of a few years ago," he said.
Totes used to provide some sort of guide to impending moves, but that too is long gone.
"Now it's just about trying to predict what price the big syndicates have things marked, as they control the market," said Lawson.
Lawson has been supplying close friends with his prices for a number of years. The fact that six different people can receive the same set of ratings and produce six completely different sets of results never ceases to amaze him and reinforces that along with the skill of doing the form, gambling is a separate skill unto itself and needs to be worked on accordingly.
More recently, Lawson has been selling his ratings successfully on a commercial basis via championbets.com.au.
"It has no impact on the market, and it's nice to have something constant coming into the bank account at the end of each month," he said.
And 'you have to speculate to accumulate'!
You can follow Nathan on Twitter @snowbet.
Nathan was a fan favourite in his time at Champion Bets, backing plenty of winners for members of his Snowy's Bets service over three years.
Since then, Mark Rhoden has taken the reigns and is doing just the same for members of NSW Winners.
Over $11,500 profit since September 2017 and there are no signs of slowing down – get on board today.