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%\journal{the Acad\'emie des sciences}
\begin{document}

\title{Motivic decomposability of generalized Severi-Brauer varieties}

%\address{Institut de Math\'ematiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France.}
\author{Maksim Zhykhovich}
\date{\today}
\maketitle


% use optional labels to link authors explicitly to addresses:
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% \address[label1]{}
% \address[label2]{}
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\selectlanguage{english}

\begin{abstract}

% Text of abstract in English
Let $F$ be an arbitrary field. Let $p$ be a positive prime number
and $D$ a central division $F$-algebra of degree $p^n$, with $n
\geq 1$. We write $SB(p^m,D)$ for the generalized Severi-Brauer
variety of right ideals in $D$ of reduced dimension $p^m$ for
$m=0,1,\ldots,n-1$. We note by $M(SB(p^m,D))$ the Chow motive with
coefficients in $\mathbb{F}_p$ of the variety $SB(p^m,D)$. It was
proven by Nikita Karpenko that this motive is indecomposable for
any prime $p$ and $m=0$ and for $p=2, m=1$ (cf. \cite{K-SB}). We
prove decomposability of $M(SB(p^m, D))$ in all the other cases
($p=2$, $1<m<n$ and $p>2$, $0<m<n$).


 %{\it To cite this article: A. Name1, A. Name2, C. R. Acad. Sci.
%Paris, Ser. I 340 (2005).}

\end{abstract}
\selectlanguage{french}
\begin{abstract}
Soient $F$ un corps arbitraire, $p$ un nombre premier positif et
$D$ une $F$-alg\`ebre de division de degr\'e $p^n$. On \'ecrit
$SB(p^m,D)$ pour la vari\'et\'e de Severi-Brauer g\'en\'eralis\'ee
des id\'eaux \`a droite de dimension r\'eduite $p^m$,
$m=0,1,\ldots,n-1$. On note par $M(SB(p^m,D))$ le motif de Chow
\`a coefficients dans $\mathbb{F}_p$ de la vari\'et\'e
$SB(p^m,D)$. Il a \'et\'e demontr\'e par Nikita Karpenko que ce
motif est indecomposable pour le nombre premier $p$ arbitraire et
$m=0$ et pour $p=2, m=1$ (cf. \cite{K-SB}). Nous prouvons la
d\'ecomposabilit\'e de $M(SB(p^m, D))$ dans tous les autres cas
($p=2$, $1<m<n$ and $p>2$, $0<m<n$).
\end{abstract}



\selectlanguage{english}
% main text
\section{Chow motives with finite coefficients}
\label{}
% etc, etc
Our basic reference for Chow groups and Chow motives (including
notations) is \cite{EKM}. We fix an associative unital commutative
ring $\Lambda$ (most frequently $\Lambda$ will be the finite field
$\mathbb{F}_p$ of $p$ elements, where $p$ is prime) and for a
variety (i.e., a separated scheme of finite type over a field) $X$
we write $\mathrm{Ch}(X)$ for its Chow group with coefficients in
$ \Lambda$ (while we write $\mathrm{CH}(X)$ for its integral Chow
group). Our category of motives is the category $\mathrm{CM}(F,
\Lambda)$ of graded Chow motives with coefficients in $\Lambda$,
\cite[definition of \S\,64]{EKM}. By a sum of motives we always
mean the direct sum. We also write $\Lambda$ for the motive
$M(\mathrm{Spec} F) \in \mathrm{CM}(F, \Lambda)$. A Tate motive is
the motive $\Lambda(i)$ with $i$ an integer.

Let $X$ be a smooth complete variety over $F$ and let $M$ be a
motive. We call $M$ split if it is a finite sum of Tate motives.
We call $X$ split, if its integral motive $M(X)\in \mathrm{CM}(F,
\mathbb{Z})$ (and therefore the motive of $X$ with an arbitrary
coefficient ring $\Lambda$) is split. We call $M$ or $X$
geometrically split, if it splits over a field extension of $F$.

Let $M$ be a geometrically split motive. Over an extension of $F$
the motive $M$ becomes isomorphic to a finite sum of Tate motives.
We write $\mathrm{rk} \, M$ and $\mathrm{rk}_i \, M$ for
respectively the number of all summands and the number of summands
$\Lambda(i)$ in this decomposition, where $i$ is an integer. Note
that these two numbers do not depend on the choice of the
splitting field extension: they are the ranks of the free
$\Lambda$-modules $\mathrm{colim}_{L/F}\, \mathrm{Ch}_{*}(M_L)$
and $\mathrm{colim}_{L/F}\, \mathrm{Ch}_{i}(M_L)$ respectively,
where the colimit is taken over all field extensions $L/F$.


 We say that $X$ satisfies the nilpotence principle, if for any
field extension $E/F$ and any coefficient ring $\Lambda$, the
kernel of the change of field homomorphism $\mathrm{End} (M(X))
\rightarrow \mathrm{End} (M(X)_E)$ consists of nilpotents. Any
projective homogeneous (under an action of a semisimple affine
algebraic group) variety is geometrically split and satisfies the
nilpotence principle, \cite[Theorem 92.4 with Remark 92.3]{EKM}.

A complete decomposition of an object in an additive category is a
finite direct sum decomposition with indecomposable summands. We
say that the Krull-Schmidt principle holds for a given object of a
given additive category, if every direct sum decomposition of the
object can be refined to a complete one (in particular, a complete
decomposition exists) and there is only one  (up to a permutation
of the summands) complete decomposition of the object. We have the
following theorem:

\theorem {\upshape(\cite[Theorem 3.6 of Chapter I]{Ch-M})}.
{Assume that the coefficient ring $\Lambda$ is finite. The
Krull-Schmidt principle holds for any shift of any summand of the
motive of any geometrically split $F$-variety satisfying the
nilpotence principle. %In other words, the Krull-Schmidt principle
%holds for the objects of the pseudo-abelian Tate subcategory in
%$CM(F,\Lambda)$ generated by the motives of the geometrically
%split $F$-varieties satisfying the nilpotence principle.}
}

\lemma \label{lem}{Assume that the coefficient ring $\Lambda$ is
finite. Let $X$ be a variety satisfying the nilpotence principle.
Let $f \in \mathrm{End} (M(X))$ and $1_E = f_E \in
\mathrm{End}(M(X)_E)$ for some field extension $E/F$. Then $f^n=1$
for some positive integer $n$.}
\proof{Since $X$ satisfies the
nilpotence principle, we have $f=1+\varepsilon$, where
$\varepsilon$ is nilpotent. Let $n$ be a positive integer such
that ${\varepsilon}^{n}=0 = n \varepsilon$. Then
$f^{n^n}=(1+\varepsilon)^{n^n}=1$ because the binomial
coefficients $n^n \choose i$ for $i<n$ are divisible by $n$.\qed }

\section{Motivic decomposability of generalized Severi-\\Brauer
varieties}

Let $p$ be a positive prime integer. The coefficient ring
$\Lambda$ is $\mathbb{F}_p$ in this section. Let $F$ be a field.
Let $D$ be a central division $F$-algebra of degree $p^n$. We
write $SB(p^m,D)$ for the generalized Severi-Brauer variety of
right ideals in $D$ of reduced dimension $p^m$ for
$m=0,1,\dots,n-1$. For the main Theorem \ref{main} we will need
the following definition.
\begin{e-definition} \label{def_tp(r,n)}{\upshape
Let $G_r(\mathbb{A}^n)$ be the Grassmann variety of $r$-planes in
$\mathbb{A}^n$. Let $c_1 = c_1(Tav) \in
\mathrm{Ch}^1(G_r(\mathbb{A}^n))$, where $Tav$ is a tautological
$r$-dimensional vector bundle on $G_r(\mathbb{A}^n)$. We define
$t_p(r,n)$ as a maximal integer $k$, such that $c_1^k \not{\!\! =}
\, 0$.}
\end{e-definition}

\begin{remark}\label{ineq_tp(r,n)}{\upshape
By \cite[Example 14.6.6]{F}, the integer $t_p(r,n)$ does not
depend on the base field and we have an inequality
$\mathrm{max}\{r, n-r  \}  \leq t_p(r,n) \leq
r(n-r)=\mathrm{dim}\, G_r(\mathbb{A}^n)$ }.
\end{remark}


\begin{lemma} \label{SB(1,D)}  Let $E/F$ be a splitting field
extension for $X=SB(1,D)$. Then the subgroup of $F$-rational
cycles in $\mathrm{Ch}_{\mathrm{dim} X }(X_E \times X_E)$ is
generated by a diagonal class.
\end{lemma}
\proof{ We write $\mathrm{\bar{Ch}}(X)$ for the image of the
homomorphism $\mathrm{Ch}(X) \rightarrow \mathrm{Ch}(X_E)$. By
\cite[Proposition 2.1.1]{K-i}, we have $\mathrm{\bar{Ch}}^i(X) =
0$ for $i>0$. Since the (say, first) projection $X^2 \rightarrow
X$ is a projective bundle, we have a (natural with respect to the
base field change) isomorphism $\mathrm{Ch}_{\mathrm{dim} \,
X}(X^2) \simeq \mathrm{Ch}(X)$. Passing to $\mathrm{\bar{Ch}}$, we
get an isomorphism $\mathrm{\bar{Ch}}_{\mathrm{dim} \, X}(X^2)
\simeq \mathrm{\bar{Ch}}(X) = \mathrm{\bar{Ch}}^0(X)$ showing that
$\mathrm{dim}_{\mathbb{F}_p}\mathrm{\bar{Ch}}_{\mathrm{dim} \,
X}(X^2)=1$. Since the diagonal class in
$\mathrm{\bar{Ch}}_{\mathrm{dim} \, X}(X^2)$ is nonzero, it
generates all the group. \qed

}

\corollary \label{M(SB(1,D))} {\upshape (cf. \cite[Theorem
2.2.1]{K-i}).}{ The motive with coefficients in $\mathbb{F}_p$ of
the Severi-Brauer variety $X=SB(1,D)$ is indecomposable. }

\proof{ To prove that our motive is indecomposable it is enough to
show that $\mathrm{End}(M(X))$= $\mathrm{Ch}_{\mathrm{dim X} }(X
\times X)$ does not contain nontrivial projectors. Let $\pi \in
\mathrm{Ch}_{\mathrm{dim} X }(X \times X)$ be a projector. By
Theorem \ref{SB(1,D)}, $\pi_E$ is zero or equal to $1_E$. Since
$X$ satisfies the nilpotence principle, $\pi$ is nilpotent in the
first case, but also idempotent, therefore $\pi$ is zero. Lemma
\ref{lem} gives us $\pi = 1$ in the second case. \qed} \\

Nikita Karpenko has recently proved the motivic indecomposability
of generalized Severi-Brauer varieties also in the case $p=2$,
$m=1$.

\begin{theorem} \label{M(SB(2,D))} {\upshape (cf. \cite[Theorem 4.2]{K-SB})}. {Let $D$ be a central division
$F$-algebra of degree $2^n$ with $n \geq 1$. Then the motive with
coefficients in $\mathbb{F}_2$ of the variety $SB(2,D)$ is
indecomposable.}
\end{theorem}

Taking into account the Corollary \ref{M(SB(1,D))}, Theorem
\ref{M(SB(2,D))} and the fact that any generalized Severi-Brauer
variety $SB(p^m,D)$ is $p$-incompressible \cite[Theorem 4.3]{K-SB}
(this condition is weaker  than motivic indecomposability), one
can expect that the Chow motive with coefficients in
$\mathbb{F}_p$ of any generalized Severi-Brauer variety
$SB(p^m,D)$ is indecomposable. But the following theorem gives us
the opposite answer.


\theorem \label{main}{Let $D$ be a central division $F$-algebra of
degree $p^n$ with $n \geq 1$. Then the motive with coefficients in
$\mathbb{F}_p$ of the variety $SB(p^m,D)$ is decomposable for
$p=2$, $1 < m <n$ and for $p>2$, $0<m<n$. In these cases
$M(SB(1,D))(k)$ is a summand of $M(SB(p^m,D))$ for $ 2 \leq k \leq
t_p(p^m,p^n)$.}

\proof{We use the notations: $X=SB(1,D),\, Y = SB(p^m,D),\,
d=\mathrm{dim}(SB(1,D))=p^n-1,\, r=p^n-p^m$. Let $E =F(X)$, then
$E/F$ is a splitting field extension for the variety $X$ (and also
for $Y$). We will show that $M(X)(k)$ is a summand of $M(Y)$. By
Lemma \ref{lem} it suffices to construct two $F$-rational
morphisms
$$ \alpha:M(X_E)(k) \rightarrow M(Y_E)\quad \mathrm{and} \quad \beta: M(Y_E) \rightarrow M(X_E)(k)  $$
satisfying $\beta \circ \alpha = 1 \in End(M(X_E)(k))=
\mathrm{Ch}_d (X_E \times X_E)$. By Theorem \ref{SB(1,D)} we can
replace condition $\beta \circ \alpha = 1$ by $\beta \circ \alpha\
\not{\! \! =}\, 0$.

Let $Tav$ be the tautological vector bundle on $X$.The product $X
\times Y$ considered over $X$ (via the first projection) is
isomorphic (as a scheme over $X$) to the Grassmann bundle
$G_r(Tav)$ of $r$-dimensional subspaces in $Tav$ (cf.
\cite[Proposition 4.3]{I-K}). Let $T$ be the tautological
$r$-dimensional vector bundle on $G_r(Tav)$. Over the field $E$
the algebra $D$ becomes isomorphic to $\mathrm{End}_{E}(V)$ for
some $E$-vector space $V$ of dimension $d+1=p^n$. We have $X_E
\simeq \mathbb{P}^{d}(V)$ and $Y_E \simeq G_{p^m}(V)$. Let $T_1$
and $T_{p^m}$ be the tautological bundles of rank $1$ and $p^m$ on
$X_E$ and $Y_E$ respectively. Then we have an isomorphism (cf.
\cite[Proposition 4.3]{I-K}): $T_E \simeq T_1 \boxtimes
(-T_{p^m})^{\vee}$ (here we lift the bundles $T_1$ and $T_{p^m}$
on $X_E \times Y_E$). Let $h=c_1(T_1)\in \mathrm{Ch}^{1}(X_E)$
(then $-h$ is a hyperplane class in $\mathrm{Ch}^{1}(X_E)$) and
$c_i=c_i({(-T_{p^m})}^{\vee})\in \mathrm{Ch}^{i}(Y_E)$. Then by
\cite[Remark 3.2.3(b)]{F}
$$ c_t(T_E)=c_t(T_1
\boxtimes (-T_{p^m})^{\vee})= \sum_{i=0}^{r}(1+(h \times
1)t)^{r-i}(1\times c_i)t^i \, .$$ It follows from the conditions
of the theorem that the binomial coefficients  ${{p^n-p^m} \choose
{2}}, {{p^n-p^m} \choose {p^m-1}}$ are divisible by $p$ and
${{p^n-p^m-1} \choose {p^m-2}} \equiv (-1)^{p^m-2}$ mod $p$.
Therefore
$$c_1(T_E) = (p^n-p^m)h\times 1 + 1 \times c_1 = 1 \times c_1 \,
,$$
$$c_2(T_E)= {{p^n-p^m} \choose {2}} h^2\times 1 + (p^n-p^m-1)h
\times c_1 + 1 \times c_2 = -h \times c_1 + 1 \times c_2 \, ,$$
\begin{multline*}
c_{p^m-1}(T_E)= {{p^n-p^m} \choose {p^m-1}} h^{p^m-1}\times 1 +
{{p^n-p^m-1} \choose {p^m-2}}h^{p^m-2}\times c_1 + \ldots =\\=
(-1)^{p^m-2}h^{p^m-2} \times c_1 + \ldots \, ,
\end{multline*}
 where ``\ldots"
stands for a linear
 combination of only those terms whose second factor has
 codimension $> 1$. For the top Chern class we have: $$c_r(T_E)=\sum_{i=0}^{r}
h^{r-i}\times c_i \, .$$ Let $\beta_1 = c_r(T_E)c_{p^m-1}(T_E)
c_2(T_E) c_1(T_E)^{k-2} =(-h)^d \times c_1^k + \ldots \, =
 x \times c_1^k + \ldots \,$, where ``\ldots" stands for a linear
 combination of only those terms whose second factor has
 codimension $> k$ and where $x$ is the class of a rational point in $\mathrm{Ch}(X_E)$.
 We take $\beta = \beta_1^{t}$, where $\beta_1^{t}$ is the transpose of $\beta_1$. Since the bundle $T$ is defined
 over $F$, the morphism $\beta \in  \mathrm{Ch}_{\mathrm{dim}Y -k}(Y_E\times X_E) = \mathrm{Hom}(M(Y_E),
 M(X_E)(k))$ is $F$-rational.

By Definition \ref{def_tp(r,n)} the cycle $c_1^k$ is non-zero. Let
$a \in \mathrm{Ch}(Y_E)$ be the element dual to $c_1^k$ with
respect to the bilinear form $\mathrm{Ch}(Y_E)\times
\mathrm{Ch}(Y_E) \rightarrow \mathbb{F}_p, \, (x_1,x_2)\mapsto
\mathrm{deg}(x_1 \cdot x_2)$. The pull-back homomorphism
$f:\mathrm{Ch}(X\times Y) \rightarrow \mathrm{Ch}(Y_{F(X)}) =
\mathrm{Ch}(Y_E)$ with respect to the morphism
$Y_{F(X)}=(\mathrm{Spec}\, F(X))\times Y \rightarrow X \times Y$
given by the generic point of $X$ is surjective by \cite[Corollary
57.11 ]{EKM}. Let $\alpha' \in \mathrm{Ch}(X \times Y)$ be a cycle
whose image in $\mathrm{Ch}(Y_E)$ under the surjection $f$ is $a$.
We define $\alpha$ as $\alpha'_{E}$ and we have $\alpha = 1 \times
a + \ldots \,$, where ``\ldots" stands for a linear combination of
only those elements whose first factor is of positive codimension
and where $1 =[X_E]$. Then $\beta \circ \alpha = 1 \times x +
\ldots \, $, where ``\ldots" stands for a linear combination of
only those terms whose first factor is of positive codimension. It
follows that $\beta \circ \alpha\ \not{\! \! =}\, 0$. \qed }

\begin{remark}
{ \upshape The Theorem \ref{main} also gives us some information
about the integral motive of the variety $SB(p^m,D)$. Indeed,
according to \cite[Corollary 2.7]{P-S-Z} the decomposition of
$M(SB(p^m,D))$ with coefficients in $\mathbb{F}_p$ lifts (and in a
unique way) to the coefficients $\mathbb{Z}/p^N \mathbb{Z}$ for
any $N \geq 2$. Then by \cite[Theorem 2.16]{P-S-Z} it lifts to
$\mathbb{Z}$ (uniquely for $p=2$ and $p=3$ and non-uniquely for
$p>3$). See also Remark \ref{intSB(4,8)}.}
\end{remark}

\begin{remark}
{\upshape Let $l$ be an integer such that $0<l<p^n$ and
$\mathrm{gcd}(l,p)=1$. The complete decomposition of the motive
$M(SB(l,D))$ with coefficients in $\mathbb{F}_p$ is described in
\cite[Proposition 2.4]{CPSZ}}.
\end{remark}

\begin{example}
{\upshape As an application of Theorem \ref{main} we describe the
complete motivic decomposition of $SB(4,D)$ for a division algebra
$D$ of degree $8$. Let $E/F$ be a splitting field extension for
the algebra $D$. We note  $ M=M(X)$. By Theorem \ref{main} and
Remark \ref{ineq_tp(r,n)}, the motives $M(2)$, $M(3)$, $M(4)$ and
by duality $M(7)$, $M(6)$, $M(5)$ are direct summands of
$M(SB(4,D))$. By \cite[Theorem 4.1]{K-SB}, we have an
indecomposable direct summand $M_{2,D}$ of $M(SB(4,D))$ with a
property: $\mathbb{F}_2(0)$ and $\mathbb{F}_2(16)$ are presented
in the decomposition of ($M_{2,D})_E$. By \cite[Theorem
3.8]{K-SB}, and Theorems \ref{M(SB(1,D))}, \ref{M(SB(2,D))} any
other indecomposable summand of $M(SB(4,D))$ is some shift of
either $M$ or $M(SB(2,D))$. But the second case is impossible
because $70 ={{8}\choose{4}}=\mathrm{rk}\, M(SB(4,D)) < 6\,
\mathrm{rk}M + \mathrm{rk}\, M(SB(2,D))= 6 \cdot 8 +
{{8}\choose{2}}= 76$.

We temporary note $X=SB(1,D), Y = SB(4,D) $. Let us assume that
the motive $M(1)$ is a summand of $M(SB(4,D))$. Then there exist
two correspondences $\alpha \in \mathrm{Ch}_{\mathrm{dim}X+1} (X
\times Y) $ and $\beta \in \mathrm{Ch}_{\mathrm{dim}Y-1} (Y \times
X)$, such that $\beta \circ \alpha=1$, where $1$ means the
diagonal class in $\mathrm{Ch}_{\mathrm{dim}X} (X \times X)$. Let
$f$ be a projection $Y \times X \rightarrow Y$. Then the cycle
$f_{*}(\beta_E) \in \mathrm{Ch}^1 (X_E)$ is $F$-rational and
non-zero. The contradiction follows from \cite[Proposition
5.1]{PicX} and \cite[Corollary 2.7]{Br(F(X))}. So $M(1)$ and by
duality $M(8)$ could not be the summands of $M(Y)$.

Assume now  that there are more than 6 motives $M$ (with some
shifts) in the decomposition of $M(SB(4,D))$. Then by duality
there are at least 8 such summands. But the decomposition of any
of these 8 summands $M$ into the sum of Tate motives over the
splitting field $E$ contains $\mathbb{F}_2(7)$. We have a
contradiction with $\mathrm{rk}_{7}\, M(SB(4,D))=7$. Therefore
\begin {equation} \label{decom}
M(SB(4,D))= M_{2,D} \oplus M(2) \oplus M(3) \oplus M(4) \oplus
M(5) \oplus M(6) \oplus M(7) \, .
\end{equation}
We can write the decomposition of $M_{2,D}$ over the function
field $L=F(SB(4,D))$:
\begin{multline*}
$$(M_{2,D})_L=\mathbb{F}_2 \oplus M(SB(1,C))(1) \oplus M(SB(2,C))(4) \oplus M(SB(2,C))(8)\\ \oplus M(SB(1,C))(12)
\oplus \mathbb{F}_2(16)\, , $$
\end{multline*}
 where $C$ is a central division
$L$-algebra (of degree $4$) Brauer-equivalent to $D_L$.
%%And we can write the decomposition of $M_{2,D}$ over the splitting
%%field $E$:
%%$$(M_{2,D})_{E}= \mathbb{F}_2(4) \oplus \mathbb{F}_2(6) \oplus \mathbb{F}_2(8) \oplus \mathbb{F}_2(10) \oplus
%%\mathbb{F}_2(12) \oplus \sum_{i=0}^{16}\mathbb{F}_2(i)\,.$$
}
\end{example}
\begin{remark} \label{intSB(4,8)}
{\upshape We have the same decomposition as (\ref{decom}) for the
integral motive of the variety $SB(4,D)$. To show this one can
apply \cite[Corollary 2.7]{P-S-Z} and then \cite[Theorem
2.16]{P-S-Z}.}
\end{remark}
{\bf Acknowledgements.} I would like to express particular
gratitude to Nikita Karpenko, my Ph.D. thesis adviser, for
introducing me to the subject, raising the question studied here,
and guiding me during this work. I am also very grateful to
Olivier Haution and Sergey Tikhonov for very useful discussions.

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\begin{thebibliography}{00}
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% the references should be in alphabetical order of authors' names.
% Articles with a single author first, author will 1 co-author next,
% then author with several co-authors;


% \bibitem{label}
% Text of bibliographic item

\bibitem{CPSZ}
Calm\`es, B., Petrov, V., Semenov, N., and Zainoulline, K. Chow
motives of twisted flag varieties. Compos. Math. 142, 4 (2006),
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\bibitem{Ch-M}
Chernousov, V., and Merkurjev,A. Motivic decomposition of
projective homogeneous varieties and the Krull-Schmidt theorem.
Transform. Groups 11, 3 (2006), 371-386.

\bibitem{PicX}
Colliot-Th\'el\`ene, J.-L., Karpenko, N., and Merkurjev, A.
Rational surfaces and canonical dimension of $\mathrm{PGL}_6$. St.
Petersburg Math. J. 19 (2008), no. 5, 793--804.


\bibitem{EKM}
Elman, R., Karpenko, N., and Merkurjev, A. {\em The algebraic and
geometric theory of quadratic forms,} vol. 56 of American
Mathematical Society Colloquium Publications. American
Mathematical Society, Providence, RI, 2008.

\bibitem{F}
Fulton, W. {\em Intersection theory}, second edition (Springer,
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\end{thebibliography}



\end{document}
