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%% Cascades of projections from log del Pezzo surfaces
%% Miles Reid and Kaori Suzuki
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 \markboth{\qquad Cascades of projections from log del Pezzo surfaces
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\title{Cascades of projections from \\ log del Pezzo surfaces}
\author{Miles Reid \and Kaori Suzuki}
\date{\em To Peter Swinnerton-Dyer, in admiration}
 \begin{document}
\maketitle

\addcontentsline{toc}{chapter}{Miles Reid and Kaori Suzuki, Cascades of
projections from log del Pezzo surfaces}
% \tableofcontents

 \begin{abstract}
 One of the best-loved tales in algebraic geometry is the saga of the blowup
of $\PP^2$ in $d\le8$ general points and its anti\-canonical embedding. If a
del Pezzo surface $F$ with log terminal singularities has a large
anti\-canonical system $|{-}K_F|$, it can likewise be blown up many times to
produce cascades of del Pezzo surfaces; as in the ancient fable, a blowup
can be viewed as a projection from a bigger weighted projective space to a
smaller one, leading in nice cases to weighted hypersurfaces or other low
codimension Gorenstein constructions. The simplest examples already give
several beautiful cascades, that we exploit as test cases for practice in
the study of various kinds of projections and unprojections. We believe that
these calculations will eventually have more serious applications to Fano
3-folds of Fano index $\ge2$, involving 1001 lovely and exotic adventures.
 \end{abstract}

 \section{The story of $\Fbar_3$} \label{sec!Fbar3}
 Once upon a time, there was a surface $F=\Fbar_3$, known to all as the
cone over the twisted cubic, or as $\PP(1,1,3)=\Proj k[u_1,u_2,v]$, where
$\wt u_1,u_2=1$, $\wt v=3$. The anti\-canonical class of $F$ is
$-K_F=\Oh_{\Fbar_3}(5)$, so that its anticanonical ring $R(F,-K_F)$ is the
fifth Veronese embedding or truncation $k[u_1,u_2,v]^{(5)}$. We see that
this ring is generated by
 \[
 \begin{array}{rcll}
 x_1,\dots,x_9&=&S^5(u_1,u_2),S^2(u_1,u_2)v & \hbox{in degree~1}, \\
 y_1,y_2&=&u_1v^3,u_2v^3 & \hbox{in degree~2}, \\
 z&=&v^5 & \hbox{in degree~3},
 \end{array}
 \]
where, as usual, we write
$S^d(u_1,u_2)=\{u_1^d,u_1^{d-1}u_2,\dots,u_2^d\}$ for the set of
monomials of degree~$d$ in $u_1,u_2$.

Note that the two generators $y_1,y_2$ in degree~2 are essential as
orbifold coordinates or {\em orbinates} at the singular point. This point
is simple and well known, but we spell it out, as it is essential for the
enjoyment of our narrative: at $P=P_v=(0,0,1)\in\PP(1,1,3)$, only $v\ne0$.
We take a cube root $\xi=\sqrt[3]v$, thus introducing a $\Z/3$ Galois
extension of the homo\-geneous coordinate ring. The homogeneous ratios
$u_1/\xi,u_2/\xi$ are coordinates on a copy of $\C^2$, which is a $\Z/3$
cover of an affine neighbourhood of $P$; hence $P$ is a quotient
singularity of type $\third(1,1)$. In our truncated subring $R(F,-K_F)$,
only $z\ne0$ at $P$, and the same orbinates are provided by the
homogeneous ratios $y_1/z^{2/3},y_2/z^{2/3}$. In the projective embedding
given by $R(F,-K_F)$, since the orbinates are naturally forms of degree~2,
we think of $P$ as a quotient singularity of type $\third(2,2)$.

There are many ways of seeing that the Hilbert function of $R(F,-K_F)$ is
given by
 \[
 P_n=h^0(F,-nK_F)= 1+\frac{25}3\binom{n+1}2 -
 \begin{cases}
 \frac{1}{3} & \hbox{if $n\equiv 1$ mod 3} \\
 0 & \hbox{otherwise}
 \end{cases}
 \]
for all $n\ge0$, and thus the Hilbert series is
 \[
 P_F(t):=\sum P_nt^n=\frac{1+7t+9t^2+7t^3+t^4}{(1-t)^2(1-t^3)}.
 \]
You can do this as an exercise in orbifold RR (\cite{YPG}, Chapter~III);
or another way is to multiply the Hilbert series $1/(1-s)^2(1-s^3)$ of
$k[u_1,u_2,v]$ through by $(1-s^5)^2(1-s^{15})$, truncate it to the
polynomial consisting only of terms of degree divisible by 5, and
substitute $s^5=t$.

Now let $S=S^{(d)}\to F$ be the blowup of $F$ in $d$ general points $P_i$,
for $d\le8$. Write $E_i$ for the $-1$-curves over $P_i$. Since
$K_S=K_F+\sum E_i$, the anticanonical ring $R(S,-K_S)$ consists of
elements of $R(F,-K_F)$ of degree~$n$ passing $n$ times through $P_i$. 
Thus each point imposes one condition in degree~1, 3 in degree~2, etc.
Therefore the Hilbert series of $S$ is
 \[
 P_S(t)=P_F(t)-d\times\frac{t}{(1-t)^3}
 =\frac{1+(7-d)t+(9-d)t^2+(7-d)t^3+t^4}{(1-t)^2(1-t^3)}.
 \]
In particular $S^{(d)}$ has anticanonical degree $\frac{25-3d}3=(8-d)+\third$.
The first cases are listed in Table~\ref{tab!ca}; the first three models
suggested by the Hilbert function work without trouble.
 \begin{table}[ht]
 \[
 \renewcommand{\arraystretch}{1.5}
 \begin{array}{|l|l|l|l|}
 \hline
 d=8 & 1/3 & P_S(t)=\frac{1+t^5}{(1-t)(1-t^2)(1-t^3)} &
 S_{10}\subset\PP(1,2,3,5) \\
 \hline
 d=7 & 4/3 & P_S(t)=\frac{1+2t^2+t^4}{(1-t)^2(1-t^3)} &
 S_{4,4}\subset\PP(1,1,2,2,3) \\
 \hline
 d=6 & 7/3 & P_S(t)=\frac{1+2t^2-2t^3-t^5}{(1-t)^3(1-t^3)} &
 S_{\Pf}\subset\PP(1^3,2^2,3) \\
 \hline
 d=5 & 10/3 & P_S(t)=\frac{1+t^2-4t^3+t^4+t^6}{(1-t)^4(1-t^3)} & \codim4 \\
 \hline
 d=4 & 13/3 & P_S(t)=\frac{1-t^2-4t^3+4t^4+t^5-t^7}{(1-t)^5(1-t^3)}
& \codim5 \\
 \hline
 \end{array}
\]
 \caption{The cascade above $S_{10}\subset\PP(1,2,3,5)$}\label{tab!ca}
 \end{table}
For $S^{(6)}$, the Hilbert function requires 3 generators in degree~1, 2
in degree~2, and 1 in degree~3, and the corresponding Hilbert numerator
is
 \[
 (1-t)^3(1-t^2)^2(1-t^3)P_S(t)=1-2t^3-3t^4+3t^5+2t^3-t^8.
 \]
This indicates that $S^{(6)}\subset\PP(1^3,2^2,3)$ should be defined (in
coordinates $x_1,x_2,x_3,y_1,y_2,z$) by the Pfaffians of a $5\times5$
skew matrix
 \begin{equation}
 A(S^{(6)})=
 \begin{pmatrix} x_1 & x_2 & b_{14} & b_{15} \\
 & x_3 & b_{24} & b_{25} \\
 && b_{34} & b_{35} \\
 &&& z
 \end{pmatrix} \quad\hbox{of degrees} \quad
 \begin{pmatrix} 1&1&2&2 \\ &1&2&2 \\ &&2&2 \\ &&&3 \end{pmatrix}.
 \label{eq!S_6}
 \end{equation}
We see that this works: thus the 3 Pfaffians involving $z$ give
$x_iz=\cdots$, so that at the point $P_z=(0,\dots,0,1)$ the three $x_i$
are eliminated as implicit functions, and $P_z$ is a $\third(2,2)$
singularity with orbinates $y_1,y_2$.

 \begin{rmk} \rm For $S^{(5)}$ and $S^{(4)}$, innocently putting in only
the generators required by the Hilbert series suggests the similar
codimension~3 Pfaffian models of Table~\ref{tab!mir}.
 \begin{table}[ht]
 \[
 \renewcommand{\arraycolsep}{1em}
 \renewcommand{\arraystretch}{2}
 \begin{array}{|l|l|l|l|} \hline
 d=5 & \frac{1-4t^3-t^4+t^4+4t^5-t^8}{(1-t)^4(1-t^2)(1-t^3)} &
 S^{(5)} \subset \PP(1^4,2,3) &
 \left(
 \begin{smallmatrix} 1&1&1&1 \\ &2&2&2 \\ &&2&2 \\ &&&2 \end{smallmatrix}
 \right) \\[6pt] \hline
 d=4 & \frac{1-t^2-4t^3+4t^4+t^5-t^7}{(1-t)^5(1-t^3)} &
 S^{(4)} \subset \PP(1^5,3) &
 \left(
 \begin{smallmatrix} 1&1&1&2 \\ &1&1&2 \\ &&1&2 \\ &&&2 \end{smallmatrix}
 \right) \\[6pt] \hline
 \end{array}
 \]
 \caption{Candidate Pfaffian models that don't work}\label{tab!mir}
 \end{table}
 However, experience says that they cannot possibly work: each of these is
a {\em mirage}\label{mirage} of a type encountered many times in the
course of previous adventures. For one thing, there is nowhere for a
variable of degree~3 to appear in the matrix, so that its Pfaffians define
a weighted projective cone with vertex $(0,\dots,0,1)$ over a base
$C\subset\PP(1^4,2)$ (respectively, $C\subset\PP^4$) that is a
projectively Gorenstein curve $C$ with $K_C=\Oh(2)$; the cone point is not
log terminal. For another, the anticanonical ring \pagebreak needs two
generators of degree~2 to provide orbinates at the singularity of type
$\third(2,2)$. The conclusion is that we have not yet put in enough
generators for the graded ring (or, in other contexts, that the variety
we seek does not exist). Mirages of this type appear all over the study
of graded rings, as discussed in \ref{ssec!mirage}.
 \end{rmk}

As we see below, $S^{(d)}$ is an explicit construction from $\Fbar_3$,
and has projections down to $S_{10}\subset\PP(1,2,3,5)$ or $S_{4,4}\subset
\PP(1,1,2,2,3)$, so that we can find out anything we want to know about the
rings $R(S,-K_S)$ by working in birational terms, either from above by
projecting from $\Fbar_3$, or from below by unprojecting from one of the
low codimension cases. We first relate without proof what happens. Listen
and attend!

Consider $S=S^{(5)}$ first. First, $R(S,-K_S)$ has two generators $y_1,y_2$
and one relation in degree~2; the Hilbert series on its own cannot detect
this, because the relation masks the second generator. Once you know about
the additional generator, the anticanonical model of $S^{(5)}$ is a
codimension~4 construction $S^{(5)}\subset\PP(1^4,2^2,3)$, with Hilbert
numerator
 \[
 (1-t)^4(1-t^2)^2(1-t^3)P_S(t)=1-t^2-4t^3+8t^5-4t^7-t^8+t^{10};
 \]
however, there is still more masking going on: although the Hilbert
series only demands one relation in degree~2 and 4 in degree~3, there
are in fact also 4 relations and 4 syzygies in degree~4, and the
ring has the $9\times16$ minimal resolution
 \begin{multline}
 \Oh_S\ot\Oh_{\PP}\ot \Oh_\PP(-2)\oplus4\Oh(-3)\oplus4\Oh(-4) \\
 \ot 4\Oh_\PP(-4)\oplus8\Oh(-5)\oplus4\Oh(-6) \ot\cdots \hbox{(sym.)}
 \label{eq!S5}
 \end{multline}
The syzygy matrixes in this complex have $4\times4$ blocks of zeros (of
degree~0). We represent this by writing out the Hilbert numerator as the
expression
 \[
 1\quad-t^2-4t^3-4t^4\quad+4t^4+8t^5+4t^6\quad-4t^6-4t^7-t^8\quad+t^{10},
 \]
where the spacing is significant. Likewise, $S^{(4)}$ is the codimension~5
construction $S^{(4)}\subset\PP(1^5,2^2,3)$, with $14\times35$ resolution
represented by
 \begin{multline}
 1\quad-3t^2-6t^3-5t^4\quad+2t^3+12t^4+15t^5+6t^6\\
 -6t^5-15t^6-12t^7-2t^8
 \quad+5t^7+6t^8+3t^9\quad-t^{11}.
 \label{eq!S4}
 \end{multline}
These assertions \pagebreak can be justified either by viewing $S^{(d)}$
as projected from $F=\Fbar_3$, or as unprojected from $S^{(d+1)}$. For
convenience, we do $S^{(5)}$ from below, and $S^{(4)}$ from above (but we
could do either case by the other method, with slightly longer
computations).

Projecting from a general $P\in S^{(5)}$ blows $P$ up to a $-1$-curve
$l=\PP^1$ contained in the Pfaffian model of
$S^{(6)}\subset\PP(1^3,2^2,3)$. Inversely, $S^{(5)}$ is obtained as the
Kustin--Miller unprojection of $l\subset S^{(6)}$ (see Papadakis and Reid
\cite{PR}): the ring of $S^{(5)}$ is generated over that of $S^{(6)}$ by
adjoining 1 unprojection variable $x=x_4$ of degree $k_S-k_l=-1-(-2)=1$,
with unprojection equations $x\cdot g_i=\cdots$, for the generators $g_i$
of $I_l$. Now $l$ is clearly a complete inter\-section of 4 hypersurfaces
of degrees $1,2,2,3$ (it is $x_3=y_1=y_2=z=0$ up to a coordinate change).
The ring of $S^{(5)}$ thus has equations the old equations of $S^{(6)}$ of
degrees $3,3,4,4,4$ (the Pfaffians (\ref{eq!S_6}) defining $A(S^{(6)})$),
together with 4 unprojection equations of degrees $2,3,3,4$. The
numerical shape of the resolution (\ref{eq!S5}) comes from this and
Gorenstein symmetry. The same result can be obtained by applying the
Kustin--Miller construction directly: the projective resolution of the
ring of $S^{(6)}$ is the Buchsbaum--Eisenbud complex $L_\bull$ of the
matrix $A(S^{(6)})$, and that of $l$ is the Koszul complex $M_\bull$ of
the regular sequence defining $l$. Then $R(S^{(5)})$ arises from a
homomorphism $L_\bull\to M_\bull$ extending the map
$\Oh_{S^{(6)}}\onto\Oh_l$. For details, see Papadakis \cite{P2}.

We justify\label{justify} $S^{(4)}$ in the other direction, by projecting down
from $F$. We can choose coordinates to put a general set of 4 points in the
form
 \[
 \{P_1,\dots,P_4\}\subset F=\PP(1,1,3) \quad\hbox{given by}\quad
 f_4(u_1,u_2)=v=0.
 \]
The anticanonical ring of the 4-point blowup $S^{(4)}$ is then generated
by
 \[
 \begin{array}{rcll}
 x_1,\dots,x_5&=&\{u_1f,u_2f,S^2(u_1,u_2)v\} & \hbox{in degree~1}, \\
 y_1,y_2&=&u_1v^3,u_2v^3 & \hbox{in degree~2}, \\
 z&=&v^5 & \hbox{in degree~3}.
 \end{array}
 \]
The ideal of relations between these can be studied by explicit
elimination (we used computer algebra, but it is not at all essential);
one finds that it is generated by
 \begin{equation}
 \renewcommand{\arraystretch}{1.3}
 \rank
 \begin{pmatrix}
 * & x_1 & x_2 & y_0 \\
 x_1 & x_3 & x_4 & y_1 \\
 x_2 & x_4 & x_5 & y_2 \\
 y_0 & y_1 & y_2 & z
 \end{pmatrix} \le1,
 \quad\hbox{where}\quad y_0=q(x_3,x_4,x_5).
 \label{eq!cod5}
 \end{equation}
Taking $y_0$ as a variable gives the second Veronese embedding of the one
point blowup of the 3-fold wps $\PP(\half,\half,\half,\frac32)$. Thus
$S^{(4)}$ is a hypersurface of weighted degree~2 in this curious weighted
quasihomogenous variety. The second Veronese embedding of the one point
blowup of $\PP^3$ is a well known codimension 5 del Pezzo variety
appearing in other myths, and its equations have a $14\times35$
resolution. We check that this agrees with (\ref{eq!S4}).

 \begin{exc}\label{exc!chron} \rm Chronicle the fate of $\Fbar_5$ and
its $d$-point blowup $S^{(d)}\to\Fbar_5$ for $d\le9$. [Hint: the Hilbert
series is 
 \begin{align*}
 P(t)&=\frac{1+9t+9t^2+11t^3+9t^4+9t^5+t^6}{(1-t)^2(1-t^5)}
 -d\times \frac{t}{(1-t)^3} \\
 &=\frac{1+(9-d)t+(9-d)t^2+(11-d)t^3+(9-d)t^4+(9-d)t^5+t^6}{(1-t)^2(1-t^5)}.
 \end{align*}
The singularity polarised by $-K=A$ is of type $\frac15(3,3)$, so that
$S^{(d)}$ is in $\PP(1^{11-d},3,3,5)$. Thus $d=9$ gives
$S_{6,6}\subset\PP(1,1,3,3,5)$ and $d=8$ gives a nice Pfaffian in
$\PP(1,1,1,3,3,5)$, with Hilbert numerator
 \[
 1-2t^4-3t^6+3t^7+2t^9-t^{13},
 \]
 etc.]

These surfaces have a singularity of type $\frac15(3,3)$; we were
disappointed at first to observe that none of these is a hyperplane
section $S\in|A|$ for a Mori Fano 3-fold $X$ of Fano index~2. For then $X$
would have a quotient singularity of type $\frac15(1,3,3)$, which is
unfortunately not terminal. For further disappointment, see \ref{ssec!not}.
 \end{exc}

 \section{The ingenious history of $\frac15(2,4)$} \label{sec!1/5(2,4)}
Let $T$ be a del Pezzo surface polarised by $-K_T=\Oh_T(A)$ with a quotient
point $P\in T$ of type $\frac15(2,4)$ as its only singularity. (Up to
isomorphism, $P$ is the quotient singularity $\frac15(1,2)$, but to give
sections of $-K_T$ weight~1, and make $\Oh_T(A)=-K_T$ the preferred generator
of the local class group, we twist $\mu_5$ by an automorphism so that
$\dd\xi\wedge\dd \eta$ is in the $\ep\mapsto\ep$ character space, and thus
$\wt \xi=2,\wt \eta=4$ mod~5.) By an exercise in the style of \cite{YPG},
Chapter~III, we see that
 \[
 P_n(T) = 1 + \binom{n+1}2A^2 -
 \begin{cases}
 0 & n\equiv0\mod5 \\
 2/5 & n\equiv1\mod5 \\
 1/5 & n\equiv2\mod5 \\
 2/5 & n\equiv3\mod5 \\
 0 & n\equiv4\mod5 \\
 \end{cases}
 \]
Trying $n=1$ gives $A^2\equiv2/5$ mod~$\Z$. Putting these values in a
Hilbert series as usual and setting $A^2=k+\frac25$ gives
 \begin{align*}
 P(t)&=\frac1{1-t}+\frac{t}{(1-t)^3}A^2
-\frac15\cdot\frac{2t+t^2+2t^3}{1-t^5}\\
 &=\frac1{1-t}+\frac{t}{(1-t)^3}k \\
 &\qquad
+\frac15\cdot\frac{2t(1+t+t^2+t^3+t^4)-(1-t)^2(2t+t^2+2t^3)}{(1-t)^2(1-t^5)}\\
 &=\frac{1-t+t^2+t^4-t^5+t^6}{(1-t)^2(1-t^5)} + \frac{t}{(1-t)^3}k.
 \end{align*}
The case $k=0$ gives
 \begin{align*}
 \frac{1-t+t^2+t^4-t^5+t^6}{(1-t)^2(1-t^5)} &=
 \frac{1+t^3+t^4+t^7}{(1-t)(1-t^2)(1-t^5)} \\
 &=\frac{1-t^6-t^8+t^{14}}{(1-t)(1-t^2)(1-t^3)(1-t^4)(1-t^5)}\,,
 \end{align*}
that is, $T_{6,8}\subset\PP(1,2,3,4,5)$.

This surface turns out to be the bottom of a cascade of six projections,
whose head is the surface $T=T_6\subset\PP(1,1,3,5)$ with $-K_T=A=\Oh(4)$.
We guessed this as follows: by the standard dimension count for del Pezzo
surfaces, we expect $T_{6,8}$ to contain a finite number of $-1$-curves
not passing through the singularity. Contracting $k$ disjoint $-1$-curves
gives a surface with $K_T^2=A^2=k+\frac25$ and the above Hilbert series.
For $k=6$, we see that $A^2=6+\frac25=\frac{32}5$ is divisible by $4^2$,
and we guess that
$A=4B$, leading to a surface with the Hilbert series of
$T=T_6\subset\PP(1,1,3,5)$. Hindsight is the only justification for this
guesswork.

One sees that the minimal resolution $\wT\to T$ is the scroll $\FF_3$ blown
up in two points on a fibre, and that $T$ is obtained from this by
contracting the chain of $\PP^1$s with self-intersection $(-3,-2)$ coming
from the negative section and the birational transform of the fibre (see
Figure~\ref{fig!resl}).
 \begin{figure}[ht]
 \begin{picture}(60,60)(-100,0)
 \put(50,55){\line(1,0){60}} \put(78,46){$-3$}
 \put(103,0){\line(0,1){60}}
 \put(106,18){$B$}
 \put(55,0){\line(0,1){60}} \put(38,42){$-2$}
 \put(50,27){\line(1,1){20}}\put(65,31){$F_1$}
 \put(50,17){\line(1,-1){20}}\put(65,7){$F_2$}
 \end{picture}
 \caption{Resolution of $T=T_6\subset\PP(1,1,3,5)$}
 \label{fig!resl}
 \end{figure}

We start by calculating the anticanonical ring of the head of the cascade,
$T=T_6\subset\PP(1,1,3,5)$. Take coordinates $u_1,u_2,v,w$ in $\PP(1,1,3,5)$,
and take the defining equation of $T$ to be
 \begin{equation}
 u_2w=f_6(u_1,v)=av^2+bvu_1^3+cu_1^6
 =l_1(v,u_1^3)l_2(v,u_1^3);
 \label{eq!l1l2}
 \end{equation}
 we could normalise the right-hand side to $(v-u_1^3)(v+u_1^3)$. We use
this relation to eliminate any monomial divisible by $u_2w$. Write $B$
for the divisor class corresponding to $\Oh_{\PP}(1)$ or its restriction
to $T$. Since $-K_T=4B$, the anticanonical embedding of $T$ is the 4th
Veronese embedding of $T\subset\PP(1,1,3,5)$; one checks that the
anticanonical ring is generated by
 \begin{equation}
 \renewcommand{\arraystretch}{1.15}
 \begin{array}{rcll}
 x_1,\dots,x_7 &=& S^4(u_1,u_2),(u_1,u_2)v & \hbox{in degree~1,} \\
 y_1,y_2 &=& u_1^3w,vw & \hbox{in degree 2,} \\
 z &=& u_1^2w^2&\hbox{in degree 3,} \\
 t &=& u_1w^3 & \hbox{in degree 4,} \\
 u &=& w^4 & \hbox{in degree 5,}
 \end{array}
 \label{eq!gens}
 \end{equation}
and that its relations are given by the $2\times2$ minors of
 \begin{equation}
 \begin{pmatrix}
 x_1&x_2&x_3&x_4&x_6&y_1&z&t \\
 x_2&x_3&x_4&x_5&x_7&A&B&C \\
 y_1&A&B&C&y_2&z&t&u
 \end{pmatrix},
 \label{eq!relns}
 \end{equation}
with
 \begin{align*}
 A&= ax_6^2+bx_1x_6+cx_1^2, \\
 B&= ax_6x_7+bx_2x_6+cx_1x_2, \\
 C&= ax_7^2+bx_3x_6+cx_1x_3.
 \end{align*}

 \begin{thm} \label{th}
 For $d\le6$, write $\si\colon T^{(d)}\broken T$ for the blowup of $T$ in
$d$ general points $P_1,\dots,P_d$. (We elucidate what ``general'' means
in (\ref{eq!general}) below.) Write $E_i$ for the $-1$-curves over $P_i$
and $A^{(d)}=\si^*A-\sum E_i$ for the anticanonical class of $T^{(d)}$.
Then $T^{(d)}$ is a log del Pezzo surface with only singularity of type
$\frac15(2,4)$ and $(-K_S)^2=6-d+\frac25$.

 For $d\le5$, the anticanonical ring of $T^{(d)}$ needs $12-d$ generators of
degrees $1^{7-d},2^2,3,4,5$, and gives an embedding
$T^{(d)}\subset\PP(1^{7-d},2^2,3,4,5)$ that takes the $E_i$ to disjoint
projectively normal lines
 \[
 E_i\iso\PP^1\subset T^{(d)}\subset\PP(1^{7-d},2^2,3,4,5).
 \]

 The anticanonical ring of $T^{(6)}$ needs $5$ generators of
degrees $1,2,3,4,5$, and embeds $T^{(6)}$ as the complete intersection
$T_{6,8}\subset\PP(1,2,3,4,5)$, taking the $E_i$ to disjoint $-1$-curves
in $T_{6,8}$ (of course, the $E_i\subset\PP(1,2,3,4,5)$ cannot be projectively
normal).
 \end{thm}

Each inclusion $R(T^{(d)},A^{(d)})\subset R(T^{(d-1)},A^{(d-1)})$ for $d\le5$
is a Kustin--Miller unprojection in the sense of \cite{PR}. That is, it
introduces precisely one new generator of degree~$1$ with pole along $E_d$,
subject only to linear relations. For $d=6$, see Remark~\ref{rmk!t2}.

 \begin{pf} As in the analogous recitations for nonsingular del Pezzo
surfaces, the proof consists for the most part of restricting to the general
curve $C\in|A^{(d)}|$. The restriction $R(T^{(d)},A^{(d)})\to R(C,A^{(d)})$
is a surjective ring homomorphism, and is the quotient by the principal ideal
$(x_C)$, where $x_C$ is the equation of $C$. Thus the hyperplane section
principle applies, and we only have to prove the appropriate generation
results for $R(C,A^{(d)})$. In the antique tale, $C$ is a nonsingular 
elliptic curve, and we win because we know everything about linear systems on
it. In our case $C\in|{-}K_{T^{(d)}}|$ is an elephant, so is again a
projectively Gorenstein curve with $K_C=0$, but it is an orbifold nodal
rational curve in a sense we are about to study. Our proof will then boil
down to a monomial calculation.

The general curve $C\in|A|$ on $T$ is irreducible and has an ordinary node at
$P$, and the two orbinates of $P\in T$ restrict to respective local analytic
coordinates on the two branches of the node. In other words, $P\in C$ is
locally analytically equivalent to the quotient
$\Bigl((\xi\eta=0)\subset\C^2\Bigr)/(\frac15(2,4))$, where $\xi,\eta$ are as
in Remark~\ref{rmk!locgen}. To make formal sense of this, we need to work
with the affine cone over $T\supset C$ along the $u$-axis, and the $\C^*$
action on them. The cone over $T$ is nonsingular along the $u$-axis outside
the origin, with transverse coordinates $y_2,t$ (see (\ref{eq!24gen})) -- the
$\frac15(2,4)$ singularity arises from the $\Z/5$ isotropy. The coefficient
of $x_6$ in the equation $(x_C)$ of $C$ is nonzero in general, corresponding
to $u_1v$ in (\ref{eq!gens}). Therefore, along the $u$-axis, the cone over
$C$ is given locally by $y_2t=\hbox{higher order terms}$.

We choose a general curve $C\in|A|$ and $d\le6$ general points
$P_1,\dots,P_d$ contained in $C$. These points are also independent general
points of $T$, because $|A|$ is a 6-dimensional linear system on $T$. This
choice ensures the existence of an irreducible curve $C\in|A-\sum P_i|$ with
the local behaviour at $P$ just described. The birational transform of $C$ on
$T^{(d)}$ is an isomorphic curve $C\in|A^{(d)}|$ that we continue to denote by
$C$. It is irreducible, therefore nef, and big since $(A^{(d)})^2>0$.

The normalisation $n\colon \wC\to C\subset T^{(d)}$ is a conventional orbifold
curve: it is a rational curve with two marked point $P_1,P_2$, the inverse
image of the node of $C$. In calculations, we take $C=\PP^1$, and $P_1=0$ and
$P_2=\infty$. It is polarised by $\wA=n^*(A^{(d)})=\frac35P_1+\frac45P_2
+(5-d)Q$, where $Q$ is some other point. This is just a notational device to
handle the sheaf of graded algebras
 \[
 \sA=\bigoplus\sA_i \quad\hbox{with}\quad \sA_i=
 \Oh_{\PP^1}\left(\left[\frac{3i}5\right]P_1+\left[\frac{4i}5\right]P_2\right)
 \tensor\Oh_{\PP^1}((5-d)i).
 \]
We calculate $R(\wC,\wA)$ in monomial terms (the answer has a nice toric
description, see Exercise~\ref{ex!JH}).

For $d=5$, the calculations is as follows: 
$R(\wC,\wA)=R(\PP^1,\frac35P_1+\frac45P_2)$ is generated by
 \begin{equation}
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{rll}
 x & \hbox{in degree 1} & \hbox{with $\div x=\frac35P_1+\frac45P_2$,} \\
 y_1,y_2 & \hbox{in degree 2} & \hbox{with $\div(y_1,y_2) = (2P_1,2P_2)+
 \frac15P_1+\frac35P_2$,} \\
 z & \hbox{in degree 3} & \hbox{with $\div z = 3P_1+
 \frac45P_1+\frac25P_2$,} \\
 t & \hbox{in degree 4} & \hbox{with $\div t = 5P_1+
 \frac25P_1+\frac15P_2$,} \\
 u_1,u_2 & \hbox{in degree 5} & \hbox{with $\div(u_1,u_2) = (7P_1,7P_2)$.}
 \end{array}
 \label{eq!1223455}
 \end{equation}
Here, in each degree, $|iD|$ is the fractional part
$\{\frac{3i}5\}P_1+\{\frac{4i}5\}P_2$ plus a linear system
$|\Oh_{\PP^1}(k_i)|$, based by elements corresponding to the monomials
$S^{k_i}(t_1,t_2)$, of which the middle ones are old, and some of the extreme
ones are new generators. Thus in degree~2, $k_2=2$, and the monomials
$y_1,x^2,y_2$ correspond to $t_1^2,t_1t_2,t_2^2$.

 \begin{exc} \rm \label{ex!JH}
 The generators of $R(\wC,\wA)$ and the relations between them are simply
grasped by noting that $u_1,t,z,y_1,x,y_2,u_2$ in (\ref{eq!1223455}) satisfy
 \[
 u_1z=t^2,\quad ty_1=z^2,\quad zx=y_1^2,\quad y_1y_2=x^4,\quad xu_2=y_2^3;
 \]
this is the Jung--Hirzebruch presentation of the invariant ring of $\Z/(35)$
acting on $\C^2$ by $\frac1{35}(1,12)$, where $[2,2,2,4,3]=\frac{35}{35-12}$.
The case $d=6$ gives $[2,2,4]=\frac{10}{7}$. Generalising this result to
the general orbifold curve $(\PP^1,\al_1P_1+\al_2P_2)$ is a little gem of
a problem.
 \end{exc}

The extension of graded rings $R(C,A^{(d)}) \subset R(\wC,\wA)$ is a
normalisation, separating two transverse sheets along the \hbox{$u$-axis}.
The affine cone over the nonnormal curve $C$ is obtained by glueing the $u_1$
and $u_2$-axes together (different choices of glueing differ by a factor in
$\C^*$, and lead to isomorphic rings). The functions compatible with this
glueing are those that take the same value on $u_1$ and $u_2$-axes. Thus
$R(C,A^{(d)}) \subset R(\wC,\wA)$ is the subring generated as above, but with
only one generator $u=u_1-u_2$ in degree~5 instead of two. This proves the
statement on generators of $R(S^{(d)},A^{(d)})$ for $d=5$. The cases
$d\le4$ are similar.

In case $d=6$, the orbifold divisor on $\wC=\PP^1$ is
 \[
 \wA=n^*(A^{(d)})=\frac35P_1+\frac45P_2-Q.
 \]
 An identical calculation shows that $R(\wC,\wA)$ is generated by
 \begin{equation}
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{rll}
 y & \hbox{in degree 2} & \hbox{with $\div y = \frac15P_1+\frac35P_2$,} \\
 z & \hbox{in degree 3} & \hbox{with $\div z = \frac45P_1+\frac25P_2$,} \\
 t & \hbox{in degree 4} & \hbox{with $\div t = P_1
 + \frac25P_1+\frac15P_2$,} \\
 u_1,u_2 & \hbox{in degree 5} & \hbox{with $\div(u_1,u_2) = (2P_1,2P_2)$.}
 \end{array}
 \label{eq!2/5}
 \end{equation}
As before, the nonnormal subring $R(C,A^{(d)})$ is generated by $y,z,t$ and
$u=u_1-u_2$, and one sees that the relations are
 \[
 yt=z^2,\quad zu=t^2-y^4.
 \]
That is, $C$ is the complete intersection $C_{6,8}\subset\PP(2,3,4,5)$, as
required.

This proves the assertion of Theorem~\ref{th} on the generation of the rings
$R(T^{(d)},A^{(d)})$. This proof uses that $A^{(d)}$ is nef and big, but not
that it is ample.

We now prove that $A^{(d)}$ is ample. It is enough to show that the
anticanonical morphism of $T^{(d)}$ does not contract any curve $\Ga$ of $T$,
or equivalently, that $T^{(d)}$ does not contain any curve with
$A^{(d)}\Ga=0$. Now because the generators of $R(T^{(d)},A^{(d)})$ include
elements $y_2,t$ in (\ref{eq!1223455}) or $y,t$ in (\ref{eq!2/5}) that give
the orbinates at $P\in A$, the anticanonical morphism of $T^{(d)}$ is an
isomorphism near $P$, and so $\Ga$ cannot pass through $P$. On the other
hand, a curve with $A^{(d)}\Ga=0$ is necessarily a component of a divisor in
the mobile linear system $|A^{(d)}|$ if $d\le5$, or $|2A^{(d)}|$ if $d=6$.

One sees that $T=T_6\subset\PP(1,1,3,5)$ has a free pencil $|B|$ defined by
$(u_1:u_2)$, with a reducible fibre $u_2=0$ that splits into two components
$F_i:(u_2=l_i=0)$, where, as in (\ref{eq!l1l2}), the equation of $T$ is
$u_2w=l_1(v,u_1^3)l_2(v,u_1^3)$ (compare Figure~\ref{fig!resl}). Every
effective Weil divisor is linearly equivalent to a positive linear combination
of $F_1,F_2$. These satisfy $F_1^2=F_2^2=-\frac25$ and $F_1F_2=\frac35$, so
that $iF_1+jF_2$ is nef if only if $\frac23j<i<\frac32j$. Moreover,
$iF_1+jF_2$ can only move away from $P$ if it is Cartier there, which happens
if and only if $5\mid(i+j)$. Next $iF_1+jF_2$ a component of
$|A|=|4F_1+4F_2|$ (resp.~$|2A|$) implies $i,j\le4$ (resp. $i,j\le8$).

Thus for $d\le5$ we just have to handle $\Ga\in|2F_1+3F_2|$ and
$|3F_1+2F_2|$. Since $-K_T\Ga=4$ and $(\Ga)^2=2$, RR gives $h^0(T,\Ga)=4$, and
for general points, no 4 of $P_1,\dots,P_d$ are contained in $\Ga$. This
completes the proof if $d\le5$. For $d=6$ we also need to consider
 \[
 4F_1+6F_2,\quad 5F_1+5F_2\quad \hbox{and}\quad 7F_1+8F_2.
 \]
The proper transform of a curve $\Ga\subset T$ will give $A^{(d)}\Ga=0$ if
$\Ga\in|A|$ passes through the $P_i$ with multiplicity $a_i$, where
 \[
 \sum a_i = -K_T\Ga, \quad \sum a_i = (\Ga)^2.
 \]
In the 3 cases above, the only solutions are
  \[
 \begin{array}{rccl}
 \Ga=4F_1+6F_2: & -K_T\Ga=8, & (\Ga)^2 =8, & \hbox{none}; \\
 \Ga=5F_1+5F_2: & -K_T\Ga=8, & (\Ga)^2 =10, &  (1,1,1,1,2,2); \\
 \Ga=7F_1+8F_2: & -K_T\Ga=12, & (\Ga)^2 =24, &  (2,2,2,2,2,2).
 \end{array}
 \]
The conclusion is that $A^{(d)}$ is ample if and only
 \begin{equation}
 \renewcommand{\arraystretch}{1.2}
 \begin{tabular}{rl}
 (0) & \hbox{the $P_i$ are distinct and contained in an irreducible curve
$C\in |A|$;} \\
 (1) & \hbox{no 4 $P_i$ are contained in any $\Ga\in|2F_1+3F_2|$ or
$|3F_1+2F_2|$;} \\
 (2) & \hbox{$|5F_1+5F_2-P_1-P_2-P_3-P_4-2P_5-2P_6|=\emptyset$;} \\
 (3) & \hbox{$|7F_1+8F_2-2\sum P_i|=\emptyset$ and $|8F_1+7F_2-2\sum
P_i|=\emptyset$.}
 \end{tabular}
 \label{eq!general}
 \end{equation}
Here conditions~(2--3) are only required if $d=6$.

These are open conditions on $P_1,\dots,P_d$, and they should fail in
codimension~1. It remains to check that they are satisfied for general
$P_1,\dots,P_6$. Write $C$ for the unique curve of $|A|$ through
$P_1,\dots,P_6$. Then any divisor $\Ga$ on $T$ in Case (2) contains $C$:
indeed,
 \[
 \bigl(5F_1+5F_2-P_1-P_2-P_3-P_4-2P_5-2P_6\bigr)\rest C
 \]
 has degree~0, but is not linear equivalent to 0 on $C$ for general
$P_1,\dots,P_6$ (recall that $C$ is a nodal cubic, so that its nonsingular
points correspond to different points of the algebraic group $\Pic C=\C^*$).
Thus $\Ga=C+B$, where $|B|=|F_1+F_2|$ is the pencil of $T$. Clearly, the
element of $|B|$ through $P_5$ does not in general pass through $P_6$. The
argument in Case~(3) is similar: a divisor $\Ga$ in Case~(3) must be of the
form $C+D$, where $D\in|3F_1+4F_2-\sum P_i|$. But
 \[
 h^0(T,3F_1+4F_2) < h^0(T,4F_1+4F_2)=h^0(T,-K_T)=7,
 \]
(see (\ref{eq!gens})) so that $|3F_1+4F_2|$ does not contain a curve through 6
general points of
$T$. \QED
 \end{pf}

 \begin{exc} \rm State and prove the analog of Theorem~\ref{th} for the
cascade of Section~\ref{sec!Fbar3}. In other words, prove that the $d$ point
blowup of $\Fbar_3$ for $d\le8$ has the properties asserted (without proof!)
throughout Section~\ref{sec!Fbar3}.
 \end{exc}

 \begin{rmk} \label{rmk!locgen} \rm
 The monomials in (\ref{eq!gens}) map to some of the local generators of the
sheaf of algebras $\bigoplus_{i=0}^4\Oh_{T,P}(i)$ at the $\frac15(2,4)$
singularity. Indeed, write $\xi,\eta$ for $\ep^2$ and $\ep^4$
eigencoordinates on $\C^2$; then $\Oh_T$ is the sheaf of invariant functions,
locally generated by $\xi^5,\xi^3\eta,\xi\eta^2,\eta^5$, whereas the
eigensheaves $\Oh_{T,P}(i)$ are modules over $\Oh_{T,P}$, and are locally
generated by
 \begin{equation}
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{lcl}
 \Oh_{T,P}(1) &\ni& \xi^3,\xi\eta,\eta^4 \\
 \Oh_{T,P}(2) &\ni& \xi,\eta^3 \\
 \Oh_{T,P}(3) &\ni& \xi^4,\xi^2\eta,\eta^2 \\
 \Oh_{T,P}(4) &\ni& \xi^2,\eta \\
 \Oh_{T,P}(5) &\ni& 1.
 \end{array}
 \label{eq!locgen}
 \end{equation}
Then the homogeneous to inhomogeneous correspondence at $P$ (setting
$\sqrt[5]w=1$) has the effect
 \[
 u_1\mapsto\eta\quad\hbox{and}\quad v\mapsto\xi,
 \]
so that the generators of $R(T,-K_T)$ map to local generators of
$\Oh_{T,P}(i)$ by
 \begin{equation}
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{ll}
 \hbox{deg 1:}& x_1=u_1^4\mapsto\eta^4, \quad x_6=u_1v\mapsto\xi\eta,
 \quad \emptyset\mapsto\xi^3; \\
 \hbox{deg 2:}& y_1=u_1^3w\mapsto\eta^3, \quad y_2=vw\mapsto\xi; \\
 \hbox{deg 3:}& z=u_1^2w^2\mapsto\eta^2, \quad x_6y_2=u_1v^2w
\mapsto\xi^2\eta, \quad \emptyset\mapsto\xi^4; \\
 \hbox{deg 4:}& t=u_1w^3\mapsto\eta, \quad y_2^2=v^2w^2\mapsto\xi^2; \\
 \hbox{deg 5:}& u=w^4\mapsto 1.
 \end{array}
 \label{eq!24gen}
 \end{equation}
The remaining generators in (\ref{eq!locgen}) are hit by monomials in these
generators: for example, $\xi^3\in\Oh_T(1)$ is first hit by $y_2^3$ in
degree~6. Thus $\Oh_T(i)$ is not always generated by its $H^0$, and not just
because the $H^0(\Oh_X(i))$ are too small. However, by ampleness, $R(T,-K_T)$
maps surjectively to local generators of $\bigoplus_{i=0}^4\Oh_T(i)$, so
that, for example, the orbinates $\xi$ and $\eta$ must be hit by some
generators of $R(T,-K_T)$.
 \end{rmk}

 \begin{rmk}[Detailed calculations of Type~II projection] \label{rmk!t2} \rm
\ We hope eventually to use the two cascades of surfaces treated in
Sections~1 and~2 as exercises in understanding Type~II unprojection as in
\cite{Ki}, Section~9, and in particular, solve the unfinished calculation
in loc.\ cit., 9.12. The unprojection from $S_{10}\subset\PP(1,2,3,5)$ to
$S_{4,4}\subset\PP(1,1,2,2,3)$ is covered by the equations of \cite{Ki}, 9.8.
The only little surprise here is that, instead of increasing the codimension
by~2, one of the entries in the $5\times5$ Pfaffian matrix is a unit, and one
of the equations masks the variable of degree~5 as a combination of other
variables.

On the other hand, the unprojection from $S_{6,8}\subset\PP(1,2,3,4,5)$ leads
to a codimension~4 ring, and the calculation is similar to the one unfinished
in \cite{Ki}, 9.12. The image $\Ga$ of $\PP^1\into\PP(1,2,3,4,5)$ cannot be
projectively normal; indeed, if $v_1,v_2$ are coordinates on $\PP^1$ and
$x,y,z,t,u$ coordinates on $\PP(1,2,3,4,5)$, the two rings have monomials
 \[
 \begin{array}{llll}
 & \PP(1,2,3,4,5) & \PP^1 \\
 \hbox{in degree 1} & x & v_1,v_2 \\
 \hbox{in degree 2} & x^2,y & v_1^2,v_1v_2,v_2^2 \\
 \hbox{in degree 3} & x^3,xy,z & v_1^3,v_1^2v_2,v_1v_2^2,v_2^3 \\
 \hbox{in degree 4} & x^4,x^2y,y^2,xz,t & S^4(v_1,v_2) \\
 \end{array}
 \]
and the restriction map from $\PP(1,2,3,4,5)$ to $\Ga$ clearly misses at least
one monomial in each degree $1,2,3$. Choose $S_{6,8}$ containing $\Ga$.
Unprojecting it adds one linear generator, and one generator in each
degree~2, 3 and~4 corresponding to these missing monomials (this will be
explained better in \cite{qG}). The old variables of degree~3 and~~4 are
masked by equations, and this gives rise to a codimension~4 surface
$S'\subset\PP(1,1,2,2,3,4,5)$. We still do not know how to complete this
calculation directly.
 \end{rmk}

 \begin{rmk} \rm
 In the projection from $\Fbar_3$ of Section~1, we always assumed that the
blown up points were in general position. In the classic epic of del Pezzo
surfaces, there are lots of interesting degenerations, most simply if 3
points in $\PP^2$ become collinear. The simplest way that blowups of
$\Fbar_3$ degenerate is that two points come to lie on a fibre $l$ of the
ruling of $\FF_3$. If we project from two points on $l$, the birational
transform of the fibre $l$ becomes a $-2$-curve, and contracting it
together with the negative section of $\FF_3$ gives a $\frac15(1,2)$
singularity. Thus all the surfaces in Section~\ref{sec!1/5(2,4)} are
degenerate projections of those in Section~\ref{sec!Fbar3}. For example,
$T_6\subset\PP(1,1,3,5)$ is a projection of $\Fbar_3$ from 2 points in a
fibre (see Figure~\ref{fig!resl}). This gives a top down elimination argument
as on page~\pageref{justify} that might allows us to complete the tricky
Type~2 unprojection calculation just discussed.

This type of contraction between surfaces with log terminal singularities
corresponds to the bad links of \cite{CPR}, 5.5. We do not make this too
precise. The fact that we blow up a point, then unexpectedly contract the
line $l$ with negative discrepancy is analogous to Sarkisov links involving
an antiflip. The regular kind of blowup of a nonsingular point in a del Pezzo
cascade decreases $K_S^2$ by 1, and the Hilbert function $P_S(t)$ by
$t/(1-t)^3=t+3t^2+6t^3+10t^4+\cdots$; whereas the special blowup (of a point
contained in a curve of degree 2/3 that is a component of a split fibre of
the conic bundle structure) considered here only decreases $K_S^2$ by 14/15,
and $P_S(t)$ by
 \begin{multline*}
 \frac{t(1+2t+3t^2+2t^3+3t^4+2t^5+t^6)}{(1-t)(1-t^3)(1-t^5)} \\
 =t+3t^2+6t^3+9t^4+14t^5+20t^6+26t^7+\cdots
 \end{multline*}
 \end{rmk}

 \section{Final remarks}

 \subsection{Why weighted projective varieties?}
Nonsingular surfaces over a field $k$ that are rational or ruled over
$\kbar$ (that is, have $\ka=-\infty$) are prominent objects of study in
birational geometry and in Diophantine geometry. By a theorem of
Castelnuovo (a distinguished precursor of Mori theory!), such a surface can
be blown down (over $k$) to a minimal surface, which is a del Pezzo
surface of rank~1, or a conic bundle over a curve with relative rank~1. In
justifying the pre-eminent position of the cubic surfaces among del Pezzo
surfaces, Peter Swinnerton-Dyer observes that del Pezzo surfaces of degree
$\ge4$ are in most respects too simple to be interesting, whereas del
Pezzo surfaces of degree~2 and~1 tend to be much too difficult. Whereas
the cubic surface is associated with the root systems $E_6$, those of
degree~2 and~1, the weighted hypersurfaces $S_4\subset\PP(1^3,2)$ and
$S_6\subset\PP(1^2,2,3)$, are associated with $E_7$ and $E_8$, and are much
more complicated from essentially every point of view (Galois theory,
biregular and birational geometry, Diophantine arithmetic, etc.). In Peter's
words:
 \begin{quote}
 ``if your research adviser gives you a problem involving del Pezzo
surfaces of degree~2 and~1, it means he really {\em hates} you.''
 \end{quote}

In view of this, working with del Pezzo surfaces with cyclic singularities
may seem perverse, since it leads to even more exotic weighted projective
constructions. For example our model case is $S_{10}\subset\PP(1,2,3,5)$, the
8 point blowup of $\Fbar_3$. It makes sense to write down the equation of
$S_{10}$ over any field, and to ask for its solutions: does this lead to any
interesting problems of birational geometry or Diophantine arithmetic? The
Galois group of the configuration of eight $-1$-curves is clearly the
symmetric groups $S_8$. In contrast to the minimal cubic surface, this is
birational over $\kbar$ in an obvious way to the conic bundle
$\FF_3\to\PP^1$, with the marked section, and a set of 8 points defined over
$k$. This suggests that our surfaces are actually simpler objects and do not
involve especially difficult or interesting Diophantine issues. On a more
positive note, log del Pezzo surfaces come in large infinite families, among
which we can surely always find some really complicated case for the graduate
student who deserves that special attention.

 \subsection{Log del Pezzo surfaces and Fano 3-folds of index~2}
\label{ssec!1/2ele}
 \subsubsection{The fabulous half-elephant}
 Our main motivation was of course to use log del Pezzo surfaces to study
Fano 3-folds of Fano index $f=2$. The {\em Fano index} of a Fano
\hbox{3-fold} $X$ in the Mori category is the maximum natural number $f$ such
that $-K_X=fA$ with $A$ a Weil divisor of $X$. Our model is the general
strategy of Alt{\i}nok, Brown and Reid \cite{ABR}, that uses K3 surfaces
as technical background and motivation in the study of Fano 3-folds of
index~1. If $X$ is a Fano with $-K_X=2A$ twice an ample Weil divisor, a
sufficiently good surface $S\in|A|$ is a del Pezzo surface (if it exists,
see below); an element of $|{-}K_X|$ is called an elephant, so $S\in|A|$
is a {\em half-elephant}. In the two cascades of Sections~1 and~2, all
the del Pezzo surfaces up to codimension~3 extend in an unobstructed way
to Fano 3-folds. Thus for example, we have Fano 3-folds of index~2
 \[
 X_{10}\subset\PP(1^2,2,3,5), \quad
 X_{4,4}\subset\PP(1^3,2^2,3), \quad\hbox{and}\quad
 X_{\Pf}\subset \PP(1^4,2^2,3)
 \]
extending the del Pezzo surfaces of Table~\ref{tab!ca}. What happens in cases
of codimension~4 is a computation based on the same projection cascade that
we have not had time to finish; the basic question is to find all Pfaffian
\hbox{3-folds} $X_{\Pf}\subset \PP(1^4,2^2,3)$ containing a linearly embedded
$\PP^2\into \PP(1^4,2^2,3)$. It seems likely that the single unprojection
type for del Pezzo surfaces from codimension~3 to 4 splits into Tom and Jerry
cases for Fano 3-folds that are essentially different (compare \cite{Ki},
Example~6.4 and~6.8 and \cite{P1}--\cite{P2}).

On the other hand, the codimension~5 surface $S^{(4)}\subset\PP(1^5,2^2,3)$ of
(\ref{eq!cod5}) probably does not have any extension in degree~1 to a Fano
3-fold of index~2: we conjecture this because it seems hard to incorporate a
new variable $x_6$ of degree~1 into the equations (\ref{eq!cod5}) in a
nontrivial way to give a 3-fold having only terminal singularities.

 \subsubsection{A good half-elephant is an extremely rare
beast}\label{ssec!not}
 In contradiction to our initial hopes, most Fano 3-folds $X$ of index~2
do not have a half-elephant, and most log del Pezzo surface $S$ do not
extend to a Fano 3-fold of index~2. An obvious necessary global condition
is $P_1(X)\ge1$, but there are also severe local restrictions on the
basket of quotient singularities: each quotient singularity
$\frac1r(1,a,r-a)$ in the basket of $X$ must have $2a\cong\pm1 \hbox{ mod
} r$ (so that when we rewrite the singularity as $\frac1r(2,2a,r-2a)$,
the equation of $S$ in degree~1 can be one of the orbinates). In slightly
different terms, as we saw in \ref{exc!chron}, a del Pezzo surface $S$
with a singularity of type $\frac1r(a,b)$, polarised by $-K_S=A$, so that
$a+b\cong1$ mod $r$, can only extend to a Fano 3-fold of index~2 if $a+1$
or $b+1\cong0$ mod $r$ (compare Example~\ref{exc!chron}), so that
$\frac1r(1,a,b)$ is terminal.

These conditions restricts the several thousand baskets for index~2 Fanos to
just a handful having a possible log del Pezzo surface as half-elephant.
 \begin{table}[ht]
 \[
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{|l|l|l|}
 \hline
1. & X_{10}\subset\PP(1,1,2,3,5) & \third(2,2,1) \\
 \hline
2. & X_{6,8}\subset\PP(1,1,2,3,4,5) & \fifth(1,2,4) \\
 \hline
3. & X_{10,14}\subset\PP(1,2,2,5,7,9) & \frac19(2,2,7) \\
 \hline
4. & X_{12,14}\subset\PP(1,2,3,4,7,11) & \frac1{11}(2,4,7) \\
 \hline
5. & X_{8,10}\subset\PP(1,2,2,3,5,7) & \third(2,2,1), \frac17(2,2,5) \\
 \hline
6. & X_{22}\subset\PP(1,2,3,7,11) & \third(2,2,1), \frac17(2,3,4) \\
 \hline
7. & X_{10,12}\subset\PP(1,2,3,4,5,9) & \third(2,2,1), \frac19(2,4,5) \\
 \hline
8. & X_{6,10}\subset\PP(1,2,2,3,5,5) & 2\times \fifth(2,2,3) \\
 \hline
9. & X_{8,12}\subset\PP(1,2,3,4,5,7) & \fifth(1,3,4), \frac17(2,3,4) \\
 \hline
10. & X_{26}\subset\PP(1,2,5,7,13) & \fifth(2,2,3), \frac17(1,2,6) \\
 \hline
11. & X_{6,8}\subset\PP(1,2,2,3,3,5) & \third(2,2,1), \fifth(2,2,3) \\
 \hline
12. & X_{10,12}\subset\PP(1,2,3,5,6,7) & 2\times \third(2,2,1), \frac17(1,2,6) \\
 \hline
13. & X_{14,18}\subset\PP(2,2,3,7,9,11) & 2\times \third(2,2,1), \frac1{11}(2,2,9)
\\
 \hline
14. & X_{8,10}\subset\PP(1,2,3,4,5,5) & \third(2,2,1), 2\times \fifth(1,2,4) \\
 \hline
15. & X_{12,14}\subset\PP(2,2,3,5,7,9) & \third(2,2,1), \fifth(2,2,3),
 \frac19(2,2,7) \\
 \hline
16. & X_{10,14}\subset\PP(2,2,3,5,7,7) & \third(2,2,1), 2\times \frac17(2,2,5) \\
 \hline
17. & X_{10,12}\subset\PP(2,2,3,5,5,7) & 2\times \fifth(2,2,3), \frac17(2,2,5) \\
 \hline
18. & X_{10,12}\subset\PP(2,3,3,4,5,7) & 4\times \third(2,2,1), \frac17(2,3,4) \\
 \hline
19. & X_{6,6}\subset\PP(1,1,2,2,3,5) & \fifth(2,2,3)\\
 \hline
 \end{array}
 \]
 \caption{Some index~2 Fano 3-folds}\label{tab!ind2}
 \end{table}
Table~\ref{tab!ind2} is a preliminary list of a few $f=2$ Fano 3-folds
without any projections from smooth points (not complete, but possibly
fairly typical). Apart from Nos.~1 and~2 that we already know from
Sections~\ref{sec!Fbar3}--\ref{sec!1/5(2,4)}, the only cases in this list
having a good half-elephant are No.~12,
$X_{10,12}\subset\PP(1,2,3,5,6,7)$ and No.~14,
$X_{8,10}\subset\PP(1,2,3,4,5,5)$.

 \subsubsection{Fano 3-folds of index~2 and projections}
 Quite independently of del Pezzo surfaces, Fano 3-folds of index~2 usually
have projections based on blowing up a nonsingular point, so often belong
to projection cascades. Suppose that $X$ is a Fano 3-fold in the Mori
category (that is, with at worst terminal singularities) and $-K_X=2A$
with $A$ a Weil divisor. Consider the blowup $\si\colon X'\to X$ at a
nonsingular point $P\in X$ with exceptional surface $E\iso\PP^2$. Then by
the adjunction formula for a blowup, $-K_{X'}=2A'$, where $A'=\si^*A-E$.
If $A^3>1$ and $P\in X$ is general then $A'$ is nef and big, and defines a
birational contraction $X'\to\Xbar$, where $\Xbar$ is again a (singular)
Fano 3-fold of index~2 containing a copy of $E\iso\PP^2$ with $\Abar\rest
E\iso\Oh_{\PP^2}(1)$; in general, $\Xbar$ will have finitely many nodes on
$E$, corresponding to the lines on $X$ through $P$. The inclusion
$R(\Xbar,\Abar)\subset R(X,A)$ is the quasi-Gorenstein unprojection of $E$
(in the sense of \cite{PR} and \cite{qG}). This means that Fano
\hbox{3-folds} of index~2 could in principle be constructed by starting
from a variety such as one of Table~\ref{tab!ind2}, force it to contain an
embedded plane $E\iso\PP^2$ of degree~1, which can then be contracted to a
nonsingular point by an unprojection. This calculation has a number of
entertaining features, not the least the question of how to describe
embeddings (say) $\PP^2\into\PP(1,2,2,5,6,9)$ and codimension~2 complete
intersections $X_{10,14}$ containing the image.

 The nonsingular case is well known: for example, a Fano 3-fold
$X\subset\PP^7$ of index~2 and degree~6 has a projection $X\broken\Xbar$,
that coincides with the linear projection from a point, whose image is a
linear section of the Grassmannian $\Grass(2,5)$ containing a linearly
embedded plane $\PP^2\subset\Xbar\into\Grass(2,5)$. There are two different
ways of embedding a plane $\PP^2\into\Grass(2,5)$ related to Schubert
conditions, and these give rise to the two families of unprojection called
Tom and Jerry, corresponding to the linear section of the Segre embedding
of the hyperplane section of $\PP^2\times\PP^2$, and
$\PP^1\times\PP^1\times\PP^2$. See \cite{P1}--\cite{P2} for details.

 \subsubsection{Alternative birational treatments} \label{sssec!Tak}
 Whereas Table~\ref{tab!ind2} (or a suitable completion), together with
unprojection of planes to nonsingular points, could thus provide a basis
for a detailed classification of Fano 3-folds of index~2 (or at least for
their numerical invariants), it is possible that many of these varieties
could be studied more easily by birational methods: in this paper we have
mainly concentrated on projections from nonsingular points, but each
projection can presumably be completed to a Sarkisov link (Corti
\cite{Co}), giving rise to a birational description.

There are alternative birational methods, for example, based on
projections from quotient singularities; these may take us outside the
Mori category, as with the ``Takeuchi program'' used by Takagi in his study of
Fano 3-folds with singular index~2 (see \cite{T}). Most of the del Pezzo
surfaces and Fano 3-folds we treat here in fact have projections of
Type~I. For example, $X_{6,8}\subset\PP(1,1,2,3,4,5)_{x_1,x_2,y,z,t,u}$
has equations
 \[
 ux_1=A_6(x_2,y,z,t) \quad\hbox{and}\quad uz=B_8(x_2,y,z,t),
 \]
 so that eliminating $u$ gives a birational map from $X_{6,8}$ to the
hypersurface
 \[
 X_9:(Bx-Az)\subset\PP(1,1,2,3,4).
 \]
Algebraically this is a Type~I projection, in fact of the simplest $Bx-Ay$
type (see \cite{Ki}, Section~2). However, from the point of view of the
Sarkisov program, it is quite different: introducing the weighted ratio
$x_2:y:t$ makes the $(1,2,4)$ blowup at $P$, not the Kawamata blowup -- it is
the blowup $X_1\to X$ with exceptional surface $E$ of discrepancy 2/5, so that
$-K_{X_1}=2(A-1/5E)$. This preserves the index~2 condition, but introduces a
line of $A_1$ singularities along the $y,t$ axis on $X_9$, taking us out of
the Mori category. Compare also Example~\ref{exa!final}.

 \subsubsection{How many Fano 3-folds of index $\ge3$ are there?}

Fano 3-folds of index $f\ge3$ do not form projection cascades -- a blowup
$X'\to X$ changes the index. Another way of seeing this is to note that
for $f\ge3$, orbifold RR applied to $\chi(-A)=0$ gives a formula for $A^3$
in terms of the basket of singularities $\sB=\{\frac1r(1,a,r-a)\}$, in much
the same what that $\frac{Ac_2}{12}$ is determined by the classic orbifold RR
formula for $\chi(\Oh_X)$:
 \[
 \frac{(-K_X)c_2}{24}=1-\sum_{\sB}\frac{r^2-1}{12r},
 \]
 (see \cite{YPG}, Corollary~10.3).

The numerical invariants of a Fano 3-fold are the data going into the
orbifold RR formula, giving the Hilbert series; compare \cite{ABR},
Section~4. It consists of $A^3$, $\frac{Ac_2}{12}$ and the basket of
singularities $\sB$; for $f\ge3$, the first two rational numbers are
determined by $\sB$.

Suzuki's Univ.\ of Tokyo thesis \cite{Su}, \cite{Su1} (based in part on
Magma programming by Gavin Brown \cite{GRD}) contains lists of the
possible numerical invariants of Fano 3-folds of index $f\ge2$. She
proves in particular that $f\le19$, with $f=19$ if and only if $X$ has
the same Hilbert series as weighted projective space $\PP(3,4,5,7)$ (we
conjecture of course that then $X\iso\PP(3,4,5,7)$.) For $f=3,\dots,19$,
the number of possible numerical types is bounded as follows:
 \[
 \renewcommand{\arraystretch}{1.2}
 \renewcommand{\arraycolsep}{0.44em}
 \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
 \hline
 f&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19 \\
 \hline
 n_f&12&9&7&1&5&3&2&0&3&0&1&0&0&0&1&0&1 \\
 \hline
 N_f&20&24&14&5&11&5&2&1\\
 \cline{1-9}
 \end{array}
 \]
Here $n_f$ is a lower bound, and $N_f$ a rough upper bound: $n_f$ refers to
the number of established cases in codimension $\le2$, that is, weighted
projective spaces, hypersurfaces or codimension~2 complete intersections.
$N_f$ is the number of candidate baskets, that includes cases in
codimension~4 and~5 that we expect to be able to justify with more work,
together with many less reputable candidates.\footnote{There are currently
some problems with the upper bound $N_f$; the rigorous bound is much larger
than given here. For details, see Suzuki's thesis \cite{Su1}.} For $f\ge9$
the number $n_f$ is correct, except for an annoying (and thoroughly
disreputable) candidate with $f=10$.

Rather remarkably, there are no codimension~3 Pfaffians except for the case
$S^{(6)}$ of Section~\ref{sec!Fbar3} (see (\ref{eq!S_6})) with $f=2$; so far
we are unable to determine which candidate cases in codimension~$\ge4$ really
occur (which accounts for the uncertainties in the list). By analogy with
Mukai's results for nonsingular Fanos, one may speculate that Fano 3-folds in
higher codimension should often be quasilinear sections of certain ``key
varieties'', such as the weighted Grassmannians treated in Corti and Reid
\cite{CR}, and there may be some convincing reason why there are few
codimension~$\ge3$ cases.

 \subsubsection{How many interesting cascades are there?}
 For present purposes, for a cascade to be of interest, at least one of the
graded rings at the bottom must be explicitly computable; for us to get some
benefit, it should realistically have codimension~$\le3$. Also, we must be
able to identify the surface at the top of the cascade, for example, because
it has higher Fano index, so is a simpler object in a Veronese embedding. The
cascades of Sections~\ref{sec!Fbar3}--\ref{sec!1/5(2,4)} illustrate how these
conditions work in ideal settings. These conditions are restrictive, and
probably only allow a small number of numerical cases. Thus, whereas each of
$\Fbar_k$ for $k=7,9,\dots$ is the head of a tall cascade, involving $k+4$
blowups, a moment's thought along the lines of Exercise~\ref{exc!chron} shows
that essentially none of the surfaces in it has anticanonical ring of small
codimension. They do not extend to Fano 3-folds of index~2 for the reason
given in Exercise~\ref{exc!chron} and \ref{ssec!not}.

As another example, consider the Fano 3-fold
$X_{10,12}\subset\PP(1,2,3,5,6,7)$ of Table~\ref{tab!ind2}, No.~12 and its
half-elephant $S_{10,12}\subset\PP(2,3,5,6,7)$. This is a surface with
quotient singularities $2\times\frac13(2,2)$ and $\frac17(2,6)$ and
$K^2=\frac2{21}$. Its minimal resolution $\wS\to S$ is a surface with
$K_{\wS}^2=-1$, so is a scroll $\FF_n$ blown up 9 times, containing two
disjoint $-3$-curves and a disjoint $-3,-2,-2$ chain of curves arising from
the $\frac17(2,6)$ singularity. $\wS$ can be constructed by blowing up
$\FF_0=\PP^1\times\PP^1$ 9 times, with 3 of the centres on each of 2 sections,
and 3 other centres infinitely near points along a nonsingular arc. It seems
likely that if these blowups are chosen generically, this surface contains
no \hbox{$-1$-curves} not passing through the singularities. Thus there seem
to be more complicated cases in which there is no cascade at all. Now, in what
way is $S_{10,12}\subset\PP(2,3,5,6,7)$ so different from
$T_{6,8}\subset\PP(1,2,3,4,5)$ of Section~\ref{sec!1/5(2,4)}?

\subsection{Mirages} \label{ssec!mirage}

Mirages have been a common phenomenon in the study of weighted pro\-jective
varieties since Fletcher's thesis. The question is to construct a graded ring
and a plausible candidate for a variety in weighted projective space having a
given Hilbert series. It happens frequently that we can find a graded ring,
but it does not correspond to a good variety, for example, because one of the
variables cannot appear in any relations for reasons of degree, so that the
candidate variety is a weighted cone. See p.~\pageref{mirage} and
Example~\ref{exa!final} below for typical cases.

A {\em mirage} is an unexpected component of a Hilbert scheme, that does not
consist of the varieties that we want, but of some degenerate cases, e.g.,
cones, varieties with index bigger than specified, or varieties condemned to
have some extra singularities. The Hilbert scheme of a family of Fano
\hbox{3-folds} may have other components, e.g., consisting of varieties with
the same numerical data, but different divisor class group. For example, the
second Veronese embedding of our index~2 Fanos $X_{10}\subset\PP(1,1,2,3,5)$
gives an extra component of the family of Fano 3-folds of index~1 with
$(-K)^3=2+\frac23$.

More generally, it is an interesting open problem to understand what these
mirages really are, and to find formal criteria to deal with them
systematically in computer generated lists. One clue is to consider how
global sections of $\Oh_X(i)$ correspond to local sections of the sheaf of
algebras $\bigoplus\Oh_{X,P}(i)$ as indicated in Remark~\ref{rmk!locgen}.

 \begin{exa} \label{exa!final} \rm
We work out one final legend that illustrates several points. Looking for
a Fano 3-fold $X$ of Fano index $f=2$ with a $\frac1{11}(2,3,8)$ terminal
quotient singularity $P\in X$ by our Hilbert series methods gives (we omit
a couple of lines of Magma)
 \[
 P_X(t)=\frac{(1-t^6)(1-t^9)(1-t^{10})}{\prod(1-t^{a_i}) : i\in
 [1,2,2,3,3,5,11]}\,.
 \]
That is, the Hilbert series of the c.i.\
$X_{6,9,10}\subset\PP(1,2,2,3,3,5,11)$. As with the examples on
p.~\pageref{mirage}, this candidate is a mirage for two reasons: the
equations cannot involve the variable of degree 11, and there is no variable
of degree 8 to act as orbinate at the singularity (this kind of thing seems
to happens fairly often with candidate models). Adding a generator of
degree~8 to the ring gives a codimension~4 model $X
\subset\PP(1,2,2,3,3,5,8,11)$. We expect that this model works: we can
eliminate the variable of degree~11 by a Type~I projection $X\broken X'$
corresponding to the $(2,3,8)$ blowup, as described in \ref{sssec!Tak}. This
weighted blowup subtracts
 \[
 \frac{t^{11}}{(1-t^2)(1-t^3)(1-t^8)(1-t^{11})}
 \]
 from $P(T)$, and a little calculation
 \begin{multline*}
 P_X(t)- \frac{t^{11}}{(1-t^2)(1-t^3)(1-t^8)(1-t^{11})} \\
 = \frac{1-t^6-t^8-t^9-t^{10}+t^{12}+t^{13}+t^{14}+t^{16}-t^{22}}
 {(1-t)(1-t^2)^2(1-t^3)^2(1-t^5)(1-t^8)}
 \end{multline*}
gives the model for the projected variety $X'$ as the Pfaffian with
weights
 \[
 \begin{pmatrix}
 1&2&3&5 \\ & 3&4&6 \\ && 5&7 \\ &&& 8
 \end{pmatrix}
 \quad\hbox{in}\quad \PP(1,2,2,3,3,5,8).
 \]
Here $X'$ is supposed to contain $\Pi=\PP(2,3,8):(x=y_1=z_1=t=0)$. The two
ways of achieving this are: take
 \[
 \left.
 \begin{array}{rl}
\hbox{Tom:} & \hbox{the first $4\times4$ block} \\
\hbox{or Jerry:} & \hbox{the first 2 rows}
 \end{array}
 \right\}
 \quad\hbox{in the ideal}\quad I_{\Pi}=(x,y_1,z_1,t),
 \]
that is, something like
 \[
 \begin{pmatrix}
 x & y_1 & z_1 & a_5 \\
 & z_1 & y_1^2 & b_6 \\
 && t & c_7 \\
 &&& d_8
 \end{pmatrix} \quad\hbox{or}\quad
 \begin{pmatrix}
 x & y_1 & z_1 & x^5 \\
 & z_1 & y_1^2 & y_1^3+z_1^2 \\
 && b'_6 & c'_7 \\
 &&& d'_8
 \end{pmatrix},
 \]
 so that $X$ can be constructed either as a Tom or a Jerry unprojection (see
\cite{PR}, \cite{P1}--\cite{P2}). As in \ref{sssec!Tak}, the projected
variety has a line of $A_1$ singularities along the $y_2,z_2$ axis.
 \end{exa}

 \begin{thebibliography}{CPR}
 \addcontentsline{toc}{section}{References}
 
 \bibitem[ABR]{ABR} S. Altn{\i}nok, G. Brown and M. Reid, Fano 3-folds, K3
surfaces and graded rings, in Singapore International Symposium in Topology
and Geometry (NUS, 2001), ed. A. J. Berrick, M. C. Leung and X. W. Xu, to
appear Contemp. Math. AMS, 2002, math.AG/0202092, 29 pp.

 \bibitem[GRD]{GRD} Gavin Brown, Graded ring database, see \\
www.maths.warwick.ac.uk/$\!\sim$gavinb/grdb.html

 \bibitem[Co]{Co} A. Corti, Factoring birational maps of threefolds after
Sarkisov, J. Algebraic Geom. {\bf4} (1995) 223--254

 \bibitem[CR]{CR} A. Corti and M. Reid, Weighted Grassmannians, in
Algebraic Geometry (Genova, Sep 2001), In memory of Paolo Francia,
M. Beltrametti and F. Catanese Eds., de Gruyter 2002, 141--163

 \bibitem[CPR]{CPR} A. Corti, A. Pukhlikov and M. Reid, Birationally rigid
Fano hypersurfaces, in Explicit birational geometry of 3-folds, A. Corti
and M. Reid (eds.), CUP 2000, 175--258

 \bibitem[Ma]{Ma} Magma (John Cannon's computer algebra system): W. Bosma,
J. Cannon and C.~Playoust, The Magma algebra system I: The user language,
J. Symb. Comp. {\bf24} (1997) 235--265. See also
www.maths.\linebreak[2]usyd.edu.au:8000/u/magma

 \bibitem[P1]{P1} Stavros Papadakis, Gorenstein rings and Kustin--Miller
unprojection, Univ. of Warwick PhD thesis, Aug 2001, pp.~vi + 72, available
from my website + Papadakis

 \bibitem[P2]{P2} Stavros Papadakis, Kustin-Miller unprojection {\em with}
complexes, J. algebraic geometry (to appear), arXiv preprint
math.AG/0111195, 23 pp. 

 \bibitem[PR]{PR} Stavros Papadakis and Miles Reid, Kustin--Miller
unprojection without complexes, J. algebraic geometry (to appear), arXiv
preprint math.AG/\linebreak[2]0011094, 15~pp.
 
 \bibitem[YPG]{YPG} Miles Reid, Young person's guide to canonical
singularities, in Algebraic Geometry (Bowdoin 1985), ed. S. Bloch, Proc.
of Symposia in Pure Math. {\bf46}, A.M.S. (1987), vol. 1, 345--414

 \bibitem[Ki]{Ki} Miles Reid, Graded rings and birational geometry, in
Proc. of algebraic geometry symposium (Kinosaki, Oct 2000), K. Ohno (Ed.),
1--72

 \bibitem[qG]{qG} Miles Reid, Quasi-Gorenstein unprojection, work in
progress, currently 17 pp.

 \bibitem[Su]{Su} Kaori Suzuki, On $\Q$-Fano 3-folds with Fano index
$\ge9$, math.AG/\linebreak[2]0210309, 7~pp.

 \bibitem[Su1]{Su1} Kaori Suzuki, On $\Q$-Fano 3-folds with Fano index
$\ge2$, Univ. of Tokyo Ph.D. thesis, 69~pp. + v, Mar~2003

 \bibitem[T]{T} TAKAGI Hiromichi, On the classification of $\Q$-Fano 3-folds
of Gorenstein index 2. I, II, RIMS preprint 1305, Nov 2000, 66 pp.

 \end{thebibliography}
\bigskip
\noindent
Miles Reid,\\
Math Inst., Univ. of Warwick,\\
Coventry CV4 7AL, England\\
e-mail: miles@maths.warwick.ac.uk \\
web: www.maths.warwick.ac.uk/$\!\sim$miles

\medskip
\noindent
SUZUKI Kaori, \\
Graduate School of Mathematical Sciences, \\
University of Tokyo \\
3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan \\
e-mail: suzuki@ms.u-tokyo.ac.jp

\end{document}
