TITLE: Complex integration and theorems
QUESTION [0 upvotes]: If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$
show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out of $C$
I could show the first part, using it
 $$f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_C \frac{f(z)}{(z-z_0)^{n+1}}$$
anyone can help in the second part?

REPLY [1 votes]: If $z_0$ lies outside $C$, then the integrand is analytic inside and on $C$. Hence, by Cauchy's theorem, the integral is $0$.


TITLE: Is it possible to combine the formula for $f_0$ and $f_{n\ge1}$ to get $f_{n\ge0}$?
QUESTION [0 upvotes]: I used generating functions to calculate the number of ways to divide a group of $n$ people into groups A, B, C and then line them up in the groups, where A must contain an odd number of people and B an even number of people. The exponential generating functions were
$$
A(x)=\frac{e^x-e^{-x}}{2}-1, \qquad B(x)=\frac{e^x+e^{-x}}{2}, \qquad C(x)=e^x.
$$
By the product rule, the number of ways to put the people in this arrangement is $F(x)=A(x)B(x)C(x)$. Thus
$$
F(x)=\frac{1}{4}\sum_{n\ge0}(3^n-2^{n+1}-(-1)^n)\frac{x^n}{n!}+\frac{1}{2},
$$
which gives me the solution:
$$
f_0=0, \qquad f_{n\ge1}=\frac{3^n-2^{n+1}-(-1)^n}{4}.
$$
Is there a way to get $f_{n\ge1}$?

REPLY [0 votes]: The problem is that $\displaystyle A(x) = \frac{e^x-e^{-x}}2$, not $\displaystyle A(x) = \frac{e^x-e^{-x}}2 - 1$. (You can tell because the $0$th coefficient, which is the number of ways to line $0$ people up in a group of odd size, should be $0$, not $-1$.) That probably fixes your issue with the coherence of the end formula.


text:
\begin{document}

\title[Generalized Laminar Families]{\large Generalized Laminar Families and \\Certain Forbidden Matrices}

\author{Peter J.~Dukes}
\address{\rm Peter J.~ Dukes:
Mathematics and Statistics,
University of Victoria, Victoria, Canada
}
\email{dukes@uvic.ca}

\thanks{Research of the author is supported by NSERC}

\date{\today}

\begin{abstract}
Recall that in a laminar family, any two sets are either disjoint or contained one in the other.  Here, a parametrized weakening of this condition is introduced.  Let us say that a set system $\cF \subseteq 2^X$ is $t$-laminar if $A,B \in \cF$ with $|A \cap B| \ge t$ implies $A \subseteq B$ or $B \subseteq A$.  We obtain very close asymptotic bounds in terms of $n$ on the maximum size of a $2$-laminar family $\cF \subseteq 2^{[n]}$.   A construction for 3-laminar families and a crude analysis for general $t$ are also given.
\end{abstract}
\maketitle
\hrule

\section{Introduction}

Let $X$ be a set.  A {\em laminar} family of subsets $\cL \subseteq 2^X$ has the property that for $A,B \in \cL$, either $A \subseteq B$, $B \subseteq A$, or $A \cap B = \emptyset$.  Consider a finite set $X$, say of cardinality $n$.  (Usually $X=[n]:=\{1,\dots,n\}$ is assumed.)  Then $|\cL| \le 2n$ by an easy (strong) induction argument.  Equality is achieved if $\cL$ consists of a full chain of size $n+1$, along with singletons.

Said another way, $\cL$ is laminar if $A,B \in \cL$ with $|A \cap B| \ge 1$, implies $A \subseteq B$ or $B \subseteq A$.  Now, let $t$ be a positive integer and say a family $\mathcal{F} \subseteq 2^X$ is $t$-\emph{laminar} if, whenever $A,B \in \cF$ with $|A \cap B| \ge t$, we have $A \subseteq B$ or $B \subseteq A$.  It is clear that an $s$-laminar family is also $t$-laminar for $t > s$.

From the point of view of obtaining upper bounds on $t$-laminar families $\cF$, there is no loss in assuming $\binom{[n]}{\le t} \subseteq \cF$.  Also, the set $[n]$ itself can be thrown into $\cF$ for free.

The author's original motivation for this topic was not to generalize laminar families.  It originated from a certain problem of Richard Anstee on forbidden configurations in zero-one matrices,  \cite{ABS} .

In this direction, say that a zero-one matrix $M$ {\em avoids} another zero-one matrix $Z$ if no permutation of rows and columns of $Z$ appears as a submatrix of $M$.  This is especially interesting when $M$ is has a fixed number $n$ of columns and both $M$ and $Z$ have no repeated rows.  (It should be clarified that Anstee works with the transpose: $m$ rows and no repeated columns, but the difference is not too important here.)

Under the assumption of distinct rows, it follows that sufficiently large $M$ are guaranteed to contain a copy of $Z$. The key question is how many rows $M$ can have while avoiding $Z$.

The rows of $M$ can be interpreted as characteristic vectors of distinct subsets of $[n]$.  Likewise, a set system $\cF \subseteq 2^{[n]}$ defines a $|\cF| \times n$ matrix $M(\cF)$ with rows indexed by $\cF$ and
$$M(\cF)_{A,i} = \begin{cases}
1 & \text{if~} i \in A \in \cF, \\
0 & \text{if~} i \not\in A.
\end{cases}
$$
Now consider the matrix
$$Z = \left[
\begin{array}{cccc}
0 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 \\
\end{array}
\right].
$$
We have chosen $Z = F_{0112}^\top$ from \cite{ABS}, where in that more general context the subscripts denote the number of rows (our columns) which are 00, 01, 10, and 11.  Incidentally, among all $2 \times 4$ matrices, the forbidden configuration problem is by far least understood for our $Z$.

Observe that $M(\cF)$ contains $Z$ as a (possibly permuted) submatrix if and only if there exist four points $w,x,y,z \in [n]$ and two sets $A,B\in \cF$ so that
\begin{itemize}
\item
$x,y,z \in A$ but $w \not\in A$; and
\item
$w,y,z \in B$ but $x \not\in B$.
\end{itemize}
It follows that forbidding $Z$ corresponds to building a $2$-laminar family.  More generally, forbidding the $2 \times (t+2)$ matrix $F_{011t}^\top$ corresponds with $t$-laminar families for each positive integer $t$.

In another direction, let us recall that $2^{[n]}$ is a graded poset.  In a laminar family, we must have that every rank one element (singleton set) belongs to a unique maximal chain in the induced ordering.  Similarly, in a $t$-laminar family, every element of rank $t$ sits below a unique maximal chain.  This is potentially a natural property for other graded posets; e.g. in the number-theoretic setting ($[n]$ with divisibility) or finite geometric setting ($\F_q^k$ with subspace containment).

Summarizing the above, we claim a collection of equivalent properties for set systems.
\begin{prop}
The following are equivalent for $\cF \subseteq 2^{[n]}$:
\vspace{-11pt}
\begin{enumerate}
\item
$\cF$ is $t$-laminar;
\item
the matrix $M(\cF)$ avoids $F_{011t}^\top$;
\item
every element of $\cF \cap \binom{[n]}{t}$ belongs to a unique maximal chain in $\cF \cap \binom{[n]}{\ge t}$.
\end{enumerate}
\end{prop}

In this note, we are especially interested in asymptotic upper bounds on such families $\cF$ for the case $t=2$ and $n \gg 0$.  We obtain fairly close estimates as our main result.

\begin{thm}
\label{main}
Let $f(n)+n+1$ be the maximum size of a $2$-laminar family on an $n$-set.  Then
\begin{eqnarray*}
\mathrm{(a)}& \displaystyle\liminf_{n \rightarrow \infty} f(n) \tbinom{n}{2}^{-1} &\ge 1.3818,~\text{and} \\
\mathrm{(b)}& \displaystyle\limsup_{n \rightarrow \infty} f(n) \tbinom{n}{2}^{-1} &\le 1.3821.
\end{eqnarray*}
\end{thm}

As the reader might expect, the lower bound follows from a constructive argument.  The upper bound is obtained from a recursive linear program.  These define Sections 2 and 3 to follow.   Interestingly, obtaining even slightly better bounds is probably very difficult; indeed, determining an exact limit as above is essentially as hard as settling certain open cases for projective planes.  We are also able to say something about the case $t=3$, but (for now) this represents a barrier.

\section{The lower bound: designs and packings}

A $t$-{\em wise balanced design} is a pair $(V,\cB)$, where $V$ is a set of $v$ {\em points} and $\cB$ is a collection of subsets of $V$ called {\em blocks} (and having size $\ge t$) such that every $t$-subset of points belongs to exactly one block.
A $t$-{\em wise packing} is defined similarly, except that some $t$-subsets of points may belong to zero blocks.

If the block sizes are a constant $k$, we speak of a $t$-$(v,k,1)$ design (or packing).  Here, the `1' denotes the (maximum) number of blocks containing a fixed $t$-subset; in more general settings this parameter may be any nonnegative integer $\lam$.

The connection with $t$-laminar families is straight from definitions.

\begin{prop}
\label{packings}
Let $\cF \subseteq 2^{[n]}$ be a $t$-laminar family.  Then any antichain in $\cF$ forms the blocks of a $t$-wise packing on $n$ points.  In particular, the maximal sets in $\cF \setminus \{[n]\}$ form a $t$-wise packing and the sets in $\cF$ of fixed cardinality $k>t$ in $\cF$ form a $t$-$(n,k,1)$ packing.
\end{prop}

A $t$-laminar family can arise from `nested' packings.

\begin{cons}
\label{nest}
Suppose $([n],\cB)$ is a $t$-wise packing.  Define $\cF$ by replacing each block $K \in \cB$ by any copy of a $t$-laminar family on $K$.
Then $\cF$ is $t$-laminar with ground set $[n]$.
\end{cons}

\begin{proof}
For $K \in \cB$, let $\cE_K$ denote the $t$-laminar family replacing $K$ in the construction, so that
$\cF = \cup_{K \in \cB} \cE_K$.  Consider $A,B \in \cF$.  If $A,B \in \cE_K$ for some $K$, it follows that $A \subseteq B$, $B \subseteq A$, or $|A \cap B| < t$.  On the other hand, if $A$ and $B$ each belong to different $\cE_{K_1}$ and $\cE_{K_2}$, then $|K_1 \cap K_2| < t$ since $\cB$ is a $t$-packing.  It follows that $|A \cap B| < t$ in this case.
\end{proof}

In the notation of our main result, Theorem~\ref{main}, recall that we are letting $f(n)$ denote the maximum $|\cF \cap \binom{[n]}{\ge 2}|$ over all 2-laminar families $\cF$.

\begin{cor}
\label{2lb}
Suppose there exists a $2$-$(n,m,1)$ packing with $b$ blocks.  Then
$f(n) \ge b \cdot f(m) + 1$.
\end{cor}

Note the `$+1$' is due to  the universal set $[n]$.
Now, Construction~\ref{nest} can be used inductively.  The choice of packings (or designs) and block sizes must now be explored further.  To this end, we recall two famous families of designs.

\begin{lemma}
\label{ap-pp}
There exists a $2$-$(q^2,q,1)$ design and a $2$-$(q^2+q+1,q+1,1)$ design for every prime power $q$.
\end{lemma}

The designs stipulated in Lemma~\ref{ap-pp} are, respectively, the affine and projective planes of order $q$.  In particular, the projective plane for $q=2$ is the well known \emph{Fano plane}, a 2-$(7,3,1)$-design.  They lead to an easy construction of $2$-laminar families for an infinite sequence of $n$.

\begin{prop}
\label{seven}
Let $n = 7^{2^r}$.  Then
\begin{equation}
\label{7bd}
f(n) \ge \binom{n}{2} \left[1+\frac{1}{\binom{3}{2}}+\frac{1}{\binom{7}{2}}+ \frac{1}{\binom{49}{2}} + \dots + \frac{1}{\binom{n}{2}} \right].
\end{equation}
\end{prop}

\begin{proof}
Use induction on $r$.  If $r=0$, then $n=7$ and we construct a family $\cF_0:= \binom{[7]}{2} \cup \cB \cup \{[7]\}$,
where $\cB \subseteq \binom{[7]}{3}$ are the $\binom{7}{2}/\binom{3}{2}$ blocks of a Fano plane.

It is easy to check that $\cF_0$ is $2$-laminar, leading to $f(7) \ge 21+7+1 = 29$.
Now for $i \ge 1$, construct $\cF_i$ from $\cF_{i-1}$ by applying Construction~\ref{nest} to the $2$-$(7^{2^i},7^{2^{i-1}},1)$ design in Lemma~\ref{ap-pp}.  It follows that $\cF_r$ is $2$-laminar with ground set $[n]$.  It remains to count $|\cF_r|$.

In general, there are $\binom{n}{2} \binom{m}{2}^{-1}$ blocks in a $2$-$(n,m,1)$ design.  With $m=7^{2^{r-1}}$ and $n=m^2$,
we have $$\frac{f(n)}{\binom{n}{2}} \ge \frac{|\cF_r|}{\binom{n}{2}} =  \frac{|\cF_{r-1}|}{\binom{m}{2}} +\frac{1}{\binom{n}{2}}.$$
We obtain the (\ref{7bd}) after an easy induction.
\end{proof}

To obtain a construction for general $n \gg 0$, we make use of an asymptotic result on maximum 2-packings.

\begin{lemma}[\cite{CCLW}]
\label{asym-pack}
Given any $k$, there exists a universal constant $c_k$ such that, for all $n$, the maximum size of a $2$-$(n,k,1)$ packing is at least $\lfloor \frac{n}{k} \lfloor \frac{n-1}{k-1} \rfloor \rfloor - c_k$.
\end{lemma}

We are now ready to prove the first half of our main result.
\begin{proof}[Proof of Theorem~\ref{main}{\rm (a)}]
Let $k=7^4$.  It follows from Proposition~\ref{seven} and a quick calculation that $f(k) \ge 1.3818 \binom{k}{2}$.  Now, take an $(n,k,1)$-packing, which by Lemma~\ref{asym-pack} has at least $b=\binom{n}{2} \binom{k}{2}^{-1}-o(n^2)$ blocks.  It follows from Corollary~\ref{2lb} that
$f(n) >  [1.3818-o(1)] \binom{n}{2}$, as desired.
\end{proof}

For $t=3$, there is a classical family of designs we can use to take the place of the finite planes.
\begin{lemma}
\label{circle}
There exists a $3$-$(q^2+1,q+1,1)$ design for every prime power $q$.
\end{lemma}
These designs are known as \emph{circle geometries}.  They arise from the action of PGL$(2,q^2)$ on $\F_{q} \cup \{\infty\}$ with its natural containment in $\F_{q^2} \cup \{\infty\}$.  Using the same construction as for $t=2$, we have the following.

\begin{prop}
\label{three}
Let $n=3^{2^r}+1$.  There exists a $3$-laminar family of $[n]$ with size
\begin{equation}
\label{3bd}
1+\binom{n}{1} + \binom{n}{2} + \binom{n}{3} \left[1+\frac{1}{\binom{4}{3}}+\frac{1}{\binom{10}{3}}+ \dots + \frac{1}{\binom{n}{2}} \right] \approx 1.5083 \binom{n}{3}.
\end{equation}
\end{prop}

Incidentally, the new result of Keevash \cite{Keevash} provides an analog of Lemma~\ref{asym-pack} for general $t$.  Thus we can expect (\ref{3bd}) as an asymptotic lower bound on $3$-laminar families for all integers $n \gg 0$, not just $n=3^{2^r}+1$.  For $t>3$, the difficult part is finding good $t$-$(n,k,1)$ packings where $n$ is not too much larger than $k$.  This is the fundamental challenge to going further.  All we can say in general is that $t$-laminar families exist with size at least $(1+\epsilon) \binom{n}{t}$, where $\epsilon>0$ depends on $t$.

\section{The upper bound: linear programming}

We begin with a structural lemma on $2$-laminar families.

\begin{lemma}
\label{struct}
Let $\cF \subseteq \binom{[n]}{\ge 2}$ be a $2$-laminar family and define $\cF^*$ to be the set of maximal elements of $\cF \setminus \{[n]\}$.  Let there be $b_i$ elements of $\cF^*$ of cardinalty $i$.  Then
$$|\cF| \le 1+\sum_{2 \le k < n} f(k) b_k.$$
\end{lemma}

\begin{proof}
Every element of $\cF \setminus \{[n]\}$ is contained in a unique element of $\cF^*$.  For each $K \in \cF^*$, the restriction of $\cF$ to sets contained in $K$ is a $2$-laminar family, say $\cF|_K$. But $|\cF|_K| \le f(|K|)$, and so the result follows by summing on $K$.
\end{proof}

Next, recall from Proposition~\ref{packings} that the maximal elements of $\cF \setminus \{[n]\}$ form a $2$-packing on $[n]$.  In this way, we can constrain the parameters $b_k$ appearing above.

\begin{lemma}
Let $([n],\cB)$ be a $2$-packing with $b_k$ blocks of size $k$, $2 \le k \le n$.  Then
\begin{equation}
\label{all-pairs}
\sum_{2 \le k \le n} \binom{k}{2} b_k \le \binom{n}{2}.
\end{equation}
Moreover, if $b_m>0$ and $b_k=0$ for $m < k \le n$, we have
\begin{equation}
\label{non-pairs}
\sum_{2 \le k \le m} \binom{k-1}{2} b_k \le \binom{n-m}{2}+\binom{m-1}{2}.
\end{equation}
\end{lemma}

\begin{proof}
The inequality (\ref{all-pairs}) follows simply by counting $\binom{[n]}{2}$ in two ways.
Next, let $B \in \cB$ with $|B|=m$ and count $\binom{[n] \setminus B}{2}$.  By the packing condition, every other block of size $k$ intersects $[n] \setminus B$ in at least $k-1$ points.  It follows that
$$\sum_{K \in \cB, K \neq B} \binom{|K|-1}{2} \le \binom{n-m}{2},$$
from which (\ref{non-pairs}) easily follows.
\end{proof}

Using just Lemma~\ref{struct} and inequality (\ref{all-pairs}), we arrive at a recursive upper bound.
\begin{prop}
\label{rec-bound}
For all $n > 2$, $$\frac{f(n)}{\binom{n}{2}} \le \frac{1}{\binom{n}{2}} + \max_{2 \le k < n} \frac{f(k)}{\binom{k}{2}}.$$
\end{prop}

\begin{proof}
$$\frac{f(n)}{\binom{n}{2}} \le \frac{1}{\binom{n}{2}} +\sum_{2 \le k < n} \frac{f(k)}{\binom{n}{2}} b_k = \frac{1}{\binom{n}{2}} + \frac{1}{\binom{n}{2}} \sum_{2 \le k < n} \frac{f(k)}{\binom{k}{2}} \cdot \binom{k}{2} b_k \le \frac{1}{\binom{n}{2}} + \max_{k < n} \frac{f(k)}{\binom{k}{2}}.\qedhere$$
\end{proof}

\begin{cor}
\label{easy-bound}
For all $n > 2$, $f(n) \le 2\binom{n}{2}$.
\end{cor}

\begin{proof}
Note that $f(3)=4$.  For $n>3$, Proposition~\ref{rec-bound} gives
$$\frac{f(n)}{\binom{n}{2}} \le \frac{f(3)}{\binom{3}{2}} + \sum_{k >3} \binom{k}{2}^{-1} = 2.\qedhere$$
\end{proof}

To use (\ref{non-pairs}), we consider $m$ as the maximum cardinality of a set in $\cF^*$.  On the one hand, larger values of $m$ contribute a larger $f(m)$ to the bound on $f(n)$.  On the other hand, larger values of $m$ constrain the packing via (\ref{non-pairs}).  To explore this trade-off, we set up a (recursive) linear program.

Define a function $\obf: \Z_{\ge 2} \rightarrow \Q$ by $\obf(2)=1$, $\obf(3) = 4$ and, for $n>3$,
$$\obf(n)=1+\max_{2 \le m < n}\LP(n,m),$$
where $\LP(n,m)$ is the solution to the linear program
\begin{equation}
\label{lp}
\begin{array}{lll}
\mbox{maximize:} & \sum_{2 \le k \le m} \obf(k) b_k \\
\mbox{subject to:} & b_m \ge 1, & b_k \ge 0 \text{ for } 3 \le k<m, \\
&(\ref{all-pairs}) \text{ and } (\ref{non-pairs}).\\
\end{array}
\end{equation}

Since this is a fractional relaxation of Lemma~\ref{struct} and the packing constraints hold, we have an upper bound on $f(n)$.
\begin{prop}
For all $n > 2$, $f(n) \le \obf(n)$.
\end{prop}

The linear program (\ref{lp}) is somewhat tricky to analyze because of the recursive dependence on $\obf$.  It is probably easiest to just appeal to a long computation which stores $\obf$ in memory.

\begin{proof}[Proof of Theorem~\ref{main}{\rm (b)}]
We compute $\obf(50000) \binom{50000}{2}^{-1} \approx 1.38206$.  Then, appealing to Proposition~\ref{struct} we have
$$f(n) \binom{n}{2}^{-1} \le 1.38206 + \sum_{k> 50000} \binom{k}{2}^{-1} \approx 1.3821. \qedhere$$
\end{proof}

It is worth providing some additional details on the computation.  First, let us dualize and slightly rewrite (\ref{lp}) so that LP$(n,m)$ is also the solution of the (now two-variable) program
\begin{equation}
\label{lp-dual}
\begin{array}{lll}
\mbox{minimize:} & \binom{n-m}{2}x + (\binom{n}{2}-\binom{m}{2}) y \\
\mbox{subject to:} & x \ge 0, y \ge 1, \text{ and}  \\
& \binom{k-1}{2} x + \binom{k}{2} y \ge \obf(k) \text{ for } k = 3,\dots,m.\\
\end{array}
\end{equation}
A major advantage now is that (\ref{lp-dual}) seems to have many redundant constraints.  For $k \ge 2$, let $\eta_k$ denote the halfspace
$$\eta_k = \left\{(x,y) : \binom{k-1}{2}x + \binom{k}{2} y \ge \obf(k) \right\}.$$
For example, $\eta_2$ is defined by $y \ge 1$.  As a special definition, let $\eta_1$ be the right halfspace defined by $x \ge 0$.
As we build a table of $\obf(n)$, we keep track of the intersection
$$\Theta_n := \bigcap_{k=1}^n \eta_k,$$ which is the feasible region for (\ref{lp-dual}).  When $n$ is incremented, it is a simple check whether $\Theta_n \subseteq \eta_{n+1}$.  If so (and this almost always happens), the constraint $\eta_{n+1}$ is eliminated.  For instance, we compute
$$\Theta_{10000} = \eta_1 \cap \eta_2 \cap \eta_3 \cap \eta_7 \cap \eta_{43} \cap \eta_{1807},$$
and we note that these subscripts (starting with 2) are obtained by iterating the map $k \mapsto k^2-k+1$.  These are the sizes of hypothetical `nested' projective planes. (Owing to the nonexistence of the $2$-$(43,7,1)$ design, though, our construction in Section 2 must shift to use powers of 7.  This is essentially what accounts for the small gap between our bounds.)  In any case, eliminating constraints at each stage reduces the calculation of LP$(n,m)$ to a minimum of evaluations of the objective function at the handful of extreme points of $\Theta_m$.  This is efficient enough for our purposes to sidestep a more rigorous analysis.
However, assuming one can prove the pattern of critical halfspaces continues, a slightly sharper upper bound than Theorem~\ref{main}(b) is possible, namely
$f(n) \binom{n}{2}^{-1} \le 1+\binom{3}{2}^{-1} + \binom{7}{2}^{-1} +\binom{43}{2}^{-1} + \dots$.

The technique in this section can, in principle, extend to $t>2$, though the bounds likely far exceed what we can construct.  A general upper bound of $O(n^t)$ on $t$-laminar families follows by a similar argument as in Corollary~\ref{easy-bound}, and that is enough for an order-of-magnitude determination.

\section*{Acknowledgements}
Thanks to Richard P.~Anstee for helpful discussions and suggestions.

config: arxiv


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\section{Examples}\mylabel{sec:examples}

In this section we give three examples of actual computations using
the localization formula applied to toric manifolds (Theorem
\ref{thm:maintheorem}). We first compute  the quantum cohomology ring
of the projective space --- this ring is of course well known, but
this makes it also a good example to experiment with our formula.
Next we compute the invariants and the quantum cohomology of the
Fano threefold $\P(\O_{\C\P^2}(2)\oplus 1)$. The reason why we
have chosen this specific example is that, as far as we know, it
is the simplest smooth projective toric variety that does not
appear in previous work, \eg. this example is not accessible by
the methods used in \cite{qr98}. Finally, we will compute some invariants
of the non--Fano threefold $\P(\O_{\C\P^2}(3)\oplus 1)$ to show that
the quantum cohomology ring of this manifold is not the one defined
in \cite{bat93}.

\subsection{The projective space}

Before we actually start computing the Gromov--Witten invariants of any complex projective
space $\P^n$ using the above formula, we will reduce the number of invariants
for which we actually have to use the formula. Remember that for Gromov--Witten invariants
the so--called composition law holds:
\[ \Phi_4^{A,X} ([pt];\alpha_1,\alpha_2,\alpha_3,\alpha_4)= \sum_{A=A_1+A_2} \sum_{i=1}^N
\Phi_3^{A_1,X}(\alpha_1,\alpha_2, \beta_i)\Phi_3^{A_2,X}(\beta^i, \alpha_3, \alpha_4), \]
where $(\beta_1,\ldots,\beta_N)$ is a basis of $H^*(X,\Z)$, $(\beta^1,\ldots, \beta^N)$ its
dual basis in the same space, and $[pt]\in\dmsp[4]$ the cohomology class Poincar\'e dual to
a point.

For the complex projective space $\P^n$, let $H\in H^2(\P^n, \Z)$
be the generator of degree--2 cohomology. Then $(1,H, H^2, \ldots, H^n)$ is a basis of $H^*(X,\Z)$
whose dual is $(H^n,\ldots,1)$. So any class $A\in H_2(\P^n, \Z)$ is necessarily a multiple of
the class $H$, $A=kH$, and if $A$ contains holomorphic curves $k$ needs to be positive. Remember
that the virtual dimension of the moduli stack $\cM_{0,3}(\P^n, A)$ is equal to
\[ \dim_\virt \cM_{0,3}(\P^n, A) = \langle c_1(\P^n), A\rangle +n + 3 -3 = k(n+1) +n. \]
Hence, for $k>1$, the virtual dimension of the moduli space is bigger than $3n$. Therefore
there can only be non--trivial Gromov--Witten invariants for the class\footnote{And of course
for the trivial class $A=0$, given by the intersection numbers (of the usual cup product).} $A=H$.

So let us look at the composition law for $A=H$. First of all, the
dimension of the (virtual) fundamental class of the moduli stack
$\M_{0,m}(\P^n, H)$ is equal to $\langle c_1(\P^n), H\rangle+ n+ m-3= 2n +m -2$. Also, we can not
decompose the class $H$ into effective classes, \ie. classes that contain again holomorphic curves.
Suppose that $p\ge q\ge r\ge 2$ and $p+q+r=2n+1$. Hence we obtain:
\begin{align*}
\Phi_3^H(H^p, H^q, H^r)&=\Phi_3^H(H^p,H^q,H^r)\cdot\Phi_3^0(H^{n-r}, H^{r-1}, H)\\
&=\Phi_4^H([pt];H^p, H^q, H^{r-1}, H) \quad \text{(since $p+q>n$)}\\
&=\Phi_4^H([pt];H^p, H^{r-1}, H^q, H) \quad \text{(since GWI are commutative)}\\
&=\Phi_3^H(H^p, H^{r-1}, H^{q+1}) \cdot \Phi_3^0(H^{n-q-1}, H^q, H)\\
&=\Phi_3^H(H^p, H^{q+1}, H^{r-1}).
\end{align*}
Therefore, by induction on $r$ we get
\[ \Phi_3^H(H^p, H^q, H^r) = \Phi_3^H(H^n, H^n, H),\]
that is we only have to use the fixed--point formula to compute one single Gromov--Witten invariant
for each complex projective space $\P^n$.

So let $(e_1,\ldots, e_n)$ be a basis of $\Z^n$, and $\Sigma$ be the fan given by the following
$1$--skeleton and set of primitive collections:
\begin{gather*}
v_1=e_1, \ldots, v_n=e_n, v_{n+1}=-e_1-\ldots-e_n\\
{\mathcal P} = \left\{\{v_1,\ldots, v_{n+1}\}\right\}.
\end{gather*}
We will denote the $n+1$ different $n$-dimensional cones in the fan $\Sigma$ as follows:
\[ \sigma_i = \langle v_1, \ldots, \hat{v_i}, \ldots, v_{n+1}\rangle, \]
where the element with the hat has to be omitted. One easily sees that the weights at $\sigma_i$
on the edge connecting to $\sigma_j$ is given by
\[ \omega_{\sigma_j}^{\sigma_i} = \we_j-\we_i.\]
As usual we will denote by $Z_1,\ldots, Z_{n+1}$ the $(\C^*)^n$--divisors in $\P^n$ coming from
the hyperplanes $\{z_i=0\}\subset\C^{n+1}$. So let us compute the invariant
$\Phi_3^H(H^n,H^n,H)$:

\begin{multline*}
\Phi_3^H(Z_1Z_2\cdots Z_n, Z_2Z_3\cdots Z_{n+1}, Z_1)
=\bbull{\sigma_{n+1}} \ml[\ \ 1] \bbull{\sigma_1}\\
=-\frac{(\we_{1}-\we_{n+1})\cdots(\we_n-\we_{n+1})\cdot(\we_2-\we_1)\cdots(\we_{n+1}-\we_1)\cdot(\we_1-\we_{n+1})}
{(\we_1-\we_{n+1})^3\cdot(\we_2-\we_{n+1})
(\we_2-\we_1)\cdots(\we_n-\we_{n+1})(\we_n-\we_1)}\\
\end{multline*}

Summing up, we get the following results for the projective space $\C\P^n$:
\begin{itemize}
\item The only non--trivial genus--zero three--point Gromov--Witten invariants are
\begin{enumerate}
\item $\Phi_3^0(H^p,H^q, H^r)=1$ if $p+q+r=n$; and
\item $\Phi_3^H(H^p,H^q, H^r)=1$ if $p+q+r=2n+1$.
\end{enumerate}
\item Its (small) quantum cohomology ring is given by
\[ QH^*(\C\P^n, \C) = \C[H, q]/_{\langle H^{n+1}-q\rangle}. \]
\end{itemize}

\subsection{The $3$--folds $\P(\O_{\P^2}(m) \oplus 1)$: some general computations}

Before we actually compute Gromov--Witten invariants of two of these projective
bundles, we do some preparatory computations for the general case, \eg. we describe
their fan, derive their moment polytope and compute their cohomology ring and weight
table.

The manifold $\P(\O_{\P^2}(m) \oplus 1)$ is $3$--dimensional, hence its fan
lives in the lattice $N:={\Z}^3$. Let $(e_1, e_2, e_3)$ be the standard basis of
${\Z}^3$. Then the one--dimensional cones of the fan $\Sigma$ that
corresponds to the manifold $\P(\O_{\P^2}(m) \oplus 1)$ are generated by the
following vectors:
\begin{alignat*}{3}
\gotv_1&=e_1&\qquad \gotv_2&=-e_1\\
\gotv_3&=e_2&\qquad\gotv_4&=e_3&\qquad\gotv_5&=-e_2-e_3-me_1.
\end{alignat*}
The set of primitive collections of the fan $\Sigma$ is given by:
\[ {\mathfrak P} = \left\{ \{ \gotv_1, \gotv_2 \}, \{ \gotv_3, \gotv_4, \gotv_5\}\right\}.
\]
In fact, it is easy to recover the projective bundle structure over the
two--dimensional projective space from this description: $\gotv_1$ and $\gotv_2$
describe the fiber, a ${\C\P}^1$, while the other three generators
describe the ${\C\P}^2$ of the base space where the twisting of
the bundle is reflected in the $(-me_1)$--term in $\gotv_5$.

From the description by the set of primitive collections,
it easy to get the set of maximal cones
in $\Sigma$. It consists of the following $3$--dimensional cones:
\begin{alignat*}{2}
\sigma_1&=\langle \gotv_1, \gotv_3, \gotv_4\rangle&\qquad
\sigma_4&=\langle \gotv_2, \gotv_3, \gotv_4\rangle\\
\sigma_2&=\langle \gotv_1, \gotv_3, \gotv_5\rangle&\qquad
\sigma_5&=\langle \gotv_2, \gotv_3, \gotv_5\rangle\\
\sigma_3&=\langle \gotv_1, \gotv_4, \gotv_5\rangle&\qquad
\sigma_6&=\langle \gotv_2, \gotv_4, \gotv_5\rangle.
\end{alignat*}

The degree--$2$ homology $H_2(X_\Sigma, {\Z})$
of $X_\Sigma=\P(\O_{\P^2}(m) \oplus 1)$ can be identified
with the group $R(\Sigma)\subset {\Z}^5$ given by
(Proposition \ref{prop:homcomtor}):
\[ R(\Sigma) := \left\{ \lambda=(\lambda^1, \ldots, \lambda^5) \in {\Z}^5
\, | \, \lambda^1\gotv_1+\cdots+\lambda^5\gotv_5=0 \right\}.
\]
The two vectors $\lambda_1:=(1,1,0,0,0)$ and
$\lambda_2=(0,-m,1,1,1)$ generate the ring $R(\Sigma)$. Moreover,
it is straightforward to check that they generate the effective
cone of $\Sigma$, \ie. the cone of homology classes that contain
holomorphic curves. Also note that by the same Proposition,
we have that
\[ \langle c_1(X_\Sigma), \lambda_1\rangle = 2 \qquad \text{and}
\qquad \langle c_1(X_\Sigma), \lambda_2\rangle = 3-m,
\]
where $c_1(X_\Sigma)$ is the first Chern class of the tangent bundle
of $X_\Sigma$.

Remember that the cohomology ring $H^*(X_\Sigma, {\C})$ is equal to the
quotient
\[ H^*(X_\Sigma, {\C}) = {\C}[Z_1, \ldots, Z_5]\left/_{
\textstyle \langle{\rm SR}(\Sigma) + {\Lin}(\Sigma)\rangle,}\right.
\]
where ${\rm SR}(\Sigma)$ is the {\em Stenley--Reisner ideal} of $\Sigma$, and
${\Lin}(\Sigma)$ is the ideal generated by linear relations. The former
is generated by monomials given by the set of primitive collections:
\[ {\rm SR}(\Sigma) = \langle Z_1 Z_2, Z_3 Z_4 Z_5 \rangle. \]
For the latter, the ideal ${\rm Lin}(\Sigma)$, let $u_1, \ldots, u_3$ be
any $\Z$--basis of the lattice $M$ dual to $N$, \eg. $u_i=e_i^*$.
Then, the ideal is generated the three terms $\sum_j \langle \gotv_j, u_i
\rangle Z_j$, \ie.
\[ Z_1-Z_2-mZ_5, \qquad Z_3-Z_5, \qquad Z_4-Z_5.\]
Therefore the cohomology ring of $X_\Sigma=\P(\O_{\P^2}(m) \oplus 1)$ is
equal to:
\begin{align*} H^*(X_\Sigma, {\C})& = {\C}[Z_1, \ldots, Z_5]\left/_{\langle
\textstyle \begin{array}{l}Z_1-Z_2-mZ_5, Z_3-Z_4,\\ Z_3-Z_5, Z_1Z_2, Z_3Z_4Z_5\end{array} \rangle}\right.\\
& = {\C}[Z_1, Z_3]\left/_{\langle
\textstyle Z_1^2-mZ_1Z_3, Z_3^3 \rangle.}\right.
\end{align*}

From the data above, it is now easy to determine the moment polytope
of $X_\Sigma$ (see figure \ref{fig:polyebdl}, and using Poincar\'e duality,
to determine the homology classes of the invariant one--dimensional irreducible
subvarieties. Note that these invariant subvarieties are mapped to the edges
of the moment polytope --- their respective homology classes are shown in
figure \ref{fig:cod1divex}.

\begin{figure}[ht]
\begin{center}
\begin{minipage}[b]{.37\linewidth}
\begin{center}
\epsfig{figure=polyebdl.eps,height=3.5cm}
\caption{The image of the moment map: the bottom triangle has side length $m\lambda_1+\lambda_2$, and the top one $\lambda_2$.}\label{fig:polyebdl}
\end{center}
\end{minipage}\hfill
\begin{minipage}[b]{72mm}
\begin{center}
\unitlength0.8mm
\begin{picture}(80,40)
\put(10,5){\circle*{2}} \put(10,35){\circle*{2}} \put(10,5){\line(0,1){30}}
\put(8,1){$\sigma_3$} \put(8,37){$\sigma_1$} \put(-5,18){$\scriptstyle m\lambda_1+\lambda_2$}
\put(70,5){\circle*{2}} \put(10,5){\line(1,0){60}} \put(70,1){$\sigma_6$} \put(38,6){$\lambda_1$}
\put(25,20){\circle*{2}} \put(10,5){\line(1,1){15}} \put(25,20){\line(-1,1){15}} \put(19,20){$\sigma_2$}
\put(17.5,10.5){$\scriptstyle m\lambda_1+\lambda_2$} \put(17.5,28){$\scriptstyle m\lambda_1+\lambda_2$}
\put(55,20){\circle*{2}} \put(25,20){\line(1,0){30}} \put(55,20){\line(1,1){15}} \put(55,20){\line(1,-1){15}}
\put(70,35){\circle*{2}} \put(70,5){\line(0,1){30}} \put(10,35){\line(1,0){60}}
\put(38,21){$\lambda_1$} \put(38,36){$\lambda_1$} \put(57,20){$\sigma_5$}
\put(62.5,13){$\scriptstyle\lambda_2$} \put(62.5,25.5){$\scriptstyle\lambda_2$}
\put(71,18){$\scriptstyle\lambda_2$}
\put(70,37){$\sigma_4$}
\end{picture}
\caption{Invariant 1--dim.\ subvarieties of the variety $\P_{\P^2}(\O(m)\oplus 1)$ and their homology classes.\newline \newline}\label{fig:cod1divex}
\end{center}
\end{minipage}
\end{center}
\end{figure}

\noindent For our calculations of the Gromov--Witten invariants, it will be convenient to have the following weight
table at hand, giving the weights of the torus action at the different charts (remember that each maximal
cone gives a chart of the toric manifold):

\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{figure}
\begin{center}
\noindent\begin{tabular}{|l@{$=$}l@{\hspace{0.3cm}}l@{$=$}l|l@{$=$}l@{\hspace{0.3cm}}l@{$=$}l|}\hline
\multicolumn{4}{|c|}{$\sigma_1=\langle v_1, v_3, v_4\rangle$}&
\multicolumn{4}{c|}{$\sigma_4=\langle v_2, v_3, v_4\rangle$}\\\hline
$u_1$&$e_1^*$&$\we_{\sigma_4}^{\sigma_1}$&$\we_{12}-m\we_5$&
$u_1$&$-e_1^*$&$\we_{\sigma_1}^{\sigma_4}$&$\we_{21}+m\we_5$\\
$u_2$&$e_2^*$&$\we_{\sigma_3}^{\sigma_1}$&$\we_3-\we_5$&
$u_2$&$e_2^*$&$\we_{\sigma_6}^{\sigma_4}$&$\we_3-\we_5$\\
$u_3$&$e_3^*$&$\we_{\sigma_2}^{\sigma_1}$&$\we_4-\we_5$&
$u_3$&$e_3^*$&$\we_{\sigma_5}^{\sigma_4}$&$\we_4-\we_5$\\\hline\hline
\multicolumn{4}{|c|}{$\sigma_2=\langle v_1, v_3, v_5\rangle$}&
\multicolumn{4}{c|}{$\sigma_5=\langle v_2, v_3, v_5\rangle$}\\\hline
$u_1$&$e_1^*-me_3^*$&$\we_{\sigma_5}^{\sigma_2}$&$\we_{12}-m\we_4$&
$u_1$&$-e_1^*+me_3^*$&$\we_{\sigma_2}^{\sigma_5}$&$\we_{21}+m\we_4$\\
$u_2$&$e_2^*-e_3^*$&$\we_{\sigma_3}^{\sigma_2}$&$\we_3-\we_4$&
$u_2$&$e_2^*-e_3^*$&$\we_{\sigma_6}^{\sigma_5}$&$\we_3-\we_4$\\
$u_3$&$-e_3^*$&$\we_{\sigma_1}^{\sigma_2}$&$\we_5-\we_4$&
$u_3$&$-e_3^*$&$\we_{\sigma_4}^{\sigma_5}$&$\we_5-\we_4$\\\hline\hline
\multicolumn{4}{|c|}{$\sigma_3=\langle v_1, v_4, v_5\rangle$}&
\multicolumn{4}{c|}{$\sigma_6=\langle v_2, v_4, v_5\rangle$}\\\hline
$u_1$&$e_1^*-me_2^*$&$\we_{\sigma_6}^{\sigma_3}$&$\we_{12}-m\we_3$&
$u_1$&$-e_1^*+me_2^*$&$\we_{\sigma_3}^{\sigma_6}$&$\we_{21}+m\we_3$\\
$u_2$&$e_3^*-e_2^*$&$\we_{\sigma_2}^{\sigma_3}$&$\we_4-\we_3$&
$u_2$&$e_3^*-e_2^*$&$\we_{\sigma_5}^{\sigma_6}$&$\we_4-\we_3$\\
$u_3$&$-e_2^*$&$\we_{\sigma_1}^{\sigma_3}$&$\we_5-\we_3$&
$u_3$&$-e_2^*$&$\we_{\sigma_4}^{\sigma_6}$&$\we_5-\we_3$\\\hline
\end{tabular}
\caption{The weight table for $\P_{P^2}(\O(m)\oplus 1)$. Substitute $\we_{12}$
by $\we_1-\we_2$ and $\we_{21}$ by $\we_2-\we_1$.}
\end{center}
\end{figure}
\end{center}

\renewcommand{\arraystretch}{1}

\subsection{The Gromov--Witten invariants of $\P(\O_{\P^2}(2) \oplus 1)$}

We will now compute all genus--$0$ three--point Gromov--Witten invariants
of the Fano threefold $\P(\O_{\P^2}(2) \oplus 1)$. The Fano property of
this manifold implies that there is only a finite number of non--zero
invariants.

We will give full details for the computations of some of the invariants,
but in some cases, due to the length of the computations, we will just give
the different graph types one need to consider and the result. However, even
those computations that are omitted here are straightforward.

Remember from above that the degree--$2$ homology classes $\lambda_1=(1,1,0,0,0)$ and
$\lambda_2=(0,-2,1,1,1)$ generate the effective cone.
Their pairing with the Chern classes $c_1(\vt)$ is two respectively one.
Any degree--$2$ homology class $A\in H_2(\vt, \Z)$ having non--zero Gromov--Witten
invariants satisfies $\langle c_1(\vt), A\rangle \le 6$; otherwise
the virtual dimension of the moduli stack were negative.

However, it is easy to see that for homology classes $A$ such that $\langle c_1(\vt, A\rangle)=6$,
all Gromov--Witten invariants are equal to zero: let $A$ be such a class, then the only
possibly non--trivial GW invariant would be
\[ \Phi^A(Z_1Z_3Z_4, Z_1Z_3Z_5, Z_1Z_4Z_5). \]
However, a graph $\mygraph$ such that the integral of these
classes over the corresponding fixed point moduli space
$\cM_\mygraph$ is non--zero has to contain the nodes $\sigma_1$,
$\sigma_2$ and $\sigma_3$. By looking at figure
\ref{fig:cod1divex} one immediately sees that such a graph
$\mygraph$ would have to have homology class $A_\mygraph$ with
$\langle c_1(\vt), A_\mygraph\rangle \geq 8$ (in this case
$A_\mygraph=3\lambda_1+2\lambda_2$). Hence, all non--zero
Gromov--Witten invariants of $\vt$ have $\langle c_1(\vt), A
\rangle \leq 5$.

We can equally exclude all classes $A=a_1\lambda_1 + a_2\lambda_2$ with $a_2>3$. For if
$\langle c_1(\vt), A \rangle \leq 5$ and $a_2>3$ we had $a_1=0$. So let us consider the
Gromov--Witten invariant
\[ \Phi^{a_2\lambda_2}(\alpha_1, \alpha_2, \alpha_3).\]
Since $\dim \vt = \langle c_1(\vt, a_2\lambda_2\rangle) = 3+
a_2>6$, at least one of the $\alpha_i$'s is of degree six, say
$\alpha_1=Z_1Z_3Z_4$. But there is no graph $\mygraph$ of homology
class a multiple of $\lambda_2$ that contains $\sigma_1$.

So we only have to compute the 3--point genus--0 Gromov--Witten invariants for the
following classes:
\[ \lambda_2, \lambda_1, 2\lambda_2, 3\lambda_2, \lambda_1+\lambda_2,
2\lambda_1, \lambda_1+2\lambda_2, \lambda_1+3\lambda_2, 2\lambda_1+\lambda_2.
\]

\begin{list}{}{}
\item[\bf $\lambda_2$--invariants:] The graph type for these invariants live on the ``upper''
triangle in the moment polytope, \ie. they can only contain the vertices $\sigma_4$, $\sigma_5$
and $\sigma_6$.
\begin{list}{}{}
\item[\bf $\Phi^{\lambda_2}(Z_1, Z_i, Z_jZ_k)=0$.] This is because the weight of $Z_1$ on the
upper triangle is zero.
\item[\bf $\Phi^{\lambda_2}(Z_i, Z_j, Z_1Z_k)=0$.] Again, the weight of $Z_1$ on the upper
triangle is zero, hence the same applies to $Z_1Z_k$.
\item[\bf $\Phi^{\lambda_2}(Z_3, Z_3,Z_4Z_5)=-1$.] In this case we get the following two possible graph types:
\[ \gtype[a] = \bbull{\sigma_4} \ml \bbull{\sigma_6}, \qquad
\gtype[b] = \bbull{\sigma_5} \ml \bbull{\sigma_6}.
\]
For both graph types, the automorphism group is trivial.
The $S_{\gtype}$ and $T_{\gtype}$ terms are as follows:
\begin{align*}
S_{\gtype[a]}&= 1^2 \cdot \frac{(\we_4-\we_3)(\we_5-\we_3)}{(\we_5-\we_3)} = (\we_4-\we_3)\\
S_{\gtype[b]}&= 1^2 \cdot \frac{(\we_4-\we_3)(\we_5-\we_3)}{(\we_4-\we_3)} = (\we_5-\we_3)\\
T_{\gtype[a]}&= \frac{(-1) \cdot 1^2}{1^2 (\we_3-\we_5)^2} \frac{(\we_{21}+\we_3\we_5)}{(\we_4-\we_5)(\we_4-\we_3)}
\cdot \frac{(\we_3-\we_5)^2}{(\we_3-\we_5)} \cdot \frac{(\we_5-\we_3)^2}{(\we_5-\we_3)}\\
&= \frac{(\we_{21} +\we_3 +\we_5)}{(\we_4-\we_5)(\we_4-\we_3)}\\
T_{\gtype[b]}&= \frac{(-1) \cdot 1^2}{1^2(\we_3-\we_4)^2} \frac{(-\we{12}+\we_3+\we_4)}{(\we_5-\we_4)(\we_4-\we_3)}
\cdot \frac{(\we_3-\we_4)^2}{(\we_3-\we_4)} \cdot \frac{(\we_4-\we_3)^2}{(\we_4-\we_3)}\\
&=\frac{(\we_{21}+\we_3+\we_4)}{(\we_5-\we_4)(\we_5-\we_3)}
\end{align*}
\end{list}
\item[\bf $\lambda_1$--invariants:] The graph types of this homology class live on the side edges,
\ie. either connect $\sigma_1$ and $\sigma_4$, $\sigma_2$ and $\sigma_5$, or $\sigma_3$ and
$\sigma_6$.
\begin{list}{}{}
\item[\bf $\Phi^{\lambda_1}(Z_i, Z_3, Z_2Z_4Z_5)=0$.] For the weight of $Z_2Z_4Z_5$ not to be zero,
the graph has to contain $\sigma_6$. However, the weight of $Z_3$ at $\sigma_3$ and $\sigma_6$
is zero.
\item[\bf $\Phi^{\lambda_1}(Z_i, Z_jZ_3, Z_4Z_5)=0$.] By a similar argument.
\item[\bf $\Phi^{\lambda_1}(Z_1, Z_1, Z_2Z_3Z_4)=1$.] The only graph type is
\[\gtype = \bbull{\sigma_1} \ml \bbull{\sigma_4}\]
Its automorphism group is trivial, and the two other terms are given by:
\begin{align*}
S_{\gtype}&=1^2\cdot(\we_3-\we_5)(\we_4-\we_5)\\
T_{\gtype}&=\frac{(-1)}{(\we_{12}-2\we_5)} \frac{1}{(\we_3-\we_5)(\we_4-\we_5)}
\frac{(\we_{12}-2\we_5)^2}{(\we_{12}-2\we_5)} \frac{(\we_{21}+2\we_5)^2}{(\we_{21}+2\we_5)}.
\end{align*}
\item[$\Phi^{\lambda_1}(Z_1, Z_1Z_4, Z_1Z_3)=1$] We get the same graph type and
the same terms $\AA_{\gtype}$, $S_{\gtype}$ and $T_{\gtype}$ as in the previous case.
\end{list}
\item[\bf $2\lambda_2$--invariants:] Again these invariants have to live on the
upper $\sigma_4$--$\sigma_5$--$\sigma_6$ triangle of the polytope, so the weight of
any class containing $Z_1$ will cause an invariant to be zero.
\begin{list}{}{}
\item[\bf $\Phi^{2\lambda_2}(Z_i, Z_j, pt)=0$.] The class $\lambda_2$ has negative self
intersection with the divisor $D_2$, hence any holomorphic curve has to lie in $D_2$,
so it can not pass trough a point in general position.
\item[\bf $\Phi^{2\lambda_2}(Z_1, Z_iZ_j, Z_kZ_l)=0$.] Simply because of $Z_1$.
\item[\bf $\Phi^{2\lambda_2}(Z_i, Z_1Z_j, Z_kZ_l)=0$.] Again because of $Z_1Z_j$.
\item[\bf $\Phi^{2\lambda_2}(Z_3, Z_3Z_4,Z_4Z_5)=-2$.] We will just give the graph types
one has to consider and leave the actual computations to the reader. The graph types
are:
\begin{alignat*}{3}
\gtype[a]&=\bbull{\sigma_4} \ml[2] \bbull{\sigma_6}&
\gtype[b]&=\bbull{\sigma_4} \ml \bbull{\sigma_6} \ml \bbull{\sigma_5}\\
\gtype[aa]&=\bbull{\sigma_4} \ml \bbull{\sigma_6} \ml \bbull{\sigma_4}&
\gtype[c]&=\bbull{\sigma_5} \ml \bbull{\sigma_4} \ml \bbull{\sigma_6} \\
\gtype[ab]&=\bbull{\sigma_6} \ml \bbull{\sigma_4} \ml \bbull{\sigma_6}&
\gtype[d]&=\bbull{\sigma_4} \ml \bbull{\sigma_5} \ml \bbull{\sigma_6}
\end{alignat*}
\end{list}
\item[\bf $3\lambda_2$--invariants:] Again, all graphs have to live on the upper
triangle of the polytope. Therefore we get:
\begin{list}{}{}
\item[\bf $\Phi^{3\lambda_2}(Z_i,Z_jZ_k,Z_1Z_lZ_p)=0$.] Because of the $Z_1$ in the third class.
\item[\bf $\Phi^{3\lambda_2}(Z_1Z_i, Z_jZ_k, Z_lZ_p)=0$.] Because of the $Z_1$ in the first class.
\item[\bf $\Phi^{3\lambda_2}(Z_3Z_4,Z_4Z_5,Z_3Z_5)=-4$.] Again, we only give the different
graph types here --- they are:
\begin{alignat*}{2}
\gtype[a]&=\bbull{\sigma_5} \ml[2] \bbull{\sigma_4} \ml \bbull{\sigma_6} &
\gtype[b]&=\bbull{\sigma_5} \ml \bbull{\sigma_4} \ml[2] \bbull{\sigma_6}\\
\gtype[aa]&=\bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6}&
\gtype[ba]&=\bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5} \mls \bbull{\sigma_4}\\
\gtype[ab]&=\fork{\sigma_5}{\sigma_4}{\sigma_6}{\sigma_5}&
\gtype[bb]&=\fork{\sigma_5}{\sigma_4}{\sigma_6}{\sigma_6}\\
\gtype[c]&=\bbull{\sigma_4} \ml[2] \bbull{\sigma_5} \ml \bbull{\sigma_6}&
\gtype[d]&=\bbull{\sigma_4} \ml \bbull{\sigma_5} \ml[2] \bbull{\sigma_6}\\
\gtype[ca]&=\bbull{\sigma_5} \mls \bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}&
\gtype[da]&=\bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}\\
\gtype[cb]&=\fork{\sigma_4}{\sigma_5}{\sigma_6}{\sigma_4}&
\gtype[db]&=\fork{\sigma_4}{\sigma_5}{\sigma_6}{\sigma_6}\\
\gtype[e]&=\bbull{\sigma_4} \ml[2] \bbull{\sigma_6} \ml \bbull{\sigma_5}&
\gtype[f]&=\bbull{\sigma_4} \ml \bbull{\sigma_6} \ml[2] \bbull{\sigma_5}\\
\gtype[ea]&=\bbull{\sigma_6} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}&
\gtype[fa]&=\bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}\\
\gtype[eb]&=\fork{\sigma_4}{\sigma_6}{\sigma_5}{\sigma_4}&
\gtype[fb]&=\fork{\sigma_4}{\sigma_5}{\sigma_5}{\sigma_5}\\
\gtype[ga]&=\bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6} \mls \bbull{\sigma_4}&
\gtype[gb]&=\bbull{\sigma_5} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}\\
\gtype[gc]&=\bbull{\sigma_6} \mls \bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}
\end{alignat*}
\end{list}
\item[\bf $(\lambda_1+\lambda_2)$--invariants:] The graph types have one component in the upper
triangle and one on a side edge --- they can not have an edge on the lower triangle
$\sigma_1$--$\sigma_2$--$\sigma_3$.
\begin{list}{}{}
\item[\bf $Phi^{\lambda_1+\lambda_2}(Z_i, Z_1Z_5,Z_1Z_3Z_4)=0$.] The third marked point has to be
at $\sigma_1$, and the second at $\sigma_2$ or $\sigma_3$ which would require an edge on the
lower triangle.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_3Z_4,Z_4Z_5,Z_3Z_5)=0$.] Needs two edges in the upper
triangle.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_1Z_3,Z_1Z_4,Z_1Z_5)=0$.] Needs an edge in the lower
triangle.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_1,Z_4Z_5,Z_1Z_3Z_4)=1$.] The only graph type to consider is
\[ \gtype = \bbull{\sigma_1} \ml \bbull{\sigma_4} \ml \bbull{\sigma_6}
\]
The automorphism group is trivial, $S_{\gtype}=(\we_4-\we_3)(\we_3-\we_5)(\we_4-\we_5)$, and
$T_{\gtype}$ the invers of $S_{\gtype}$.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_3,Z_4Z_5,Z_1Z_3Z_4)=1$.] Same graph and terms
as in the previous case.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_1Z_3,Z_4Z_5,Z_3Z_4)=1$.] Again the same.
\item[\bf $\Phi^{\lambda_1+\lambda_2}(Z_1Z_3,Z_4Z_5,Z_1Z_4)=1$.] Again the same.
\end{list}
\item[\bf $2\lambda_1$--invariants are all zero:]
The virtual dimension of the corresponding moduli space is seven, so for the
invariants $\Phi^{2\lambda_1}(\alpha_1,\alpha_2,\alpha_3)$ we have two cases for
the classes $\alpha_i$:
\begin{itemize}
\item $\deg \alpha_1=\deg \alpha_2=6$ and $\deg \alpha_3=2$\newline
In this cases we can set $\alpha_1=Z_1Z_3Z_4$ and
$\alpha_2=Z_1Z_4Z_5$. There is obviously no graph $\mygraph$ in
the homology class $2\lambda_1$ containing both $\sigma_1$ and
$\sigma_3$.
\item $\deg \alpha_1=6$ and $\deg \alpha_2=\deg \alpha_3=4$\newline
We can set $\alpha_1=Z_1Z_3Z_4$. For $\alpha_2$ we have to
choices: $\alpha_2=Z_4Z_5$ or $\alpha_2=Z_1Z_5$. Again, there is
no graph $\mygraph$ with homology class $2\lambda_1$ that contains
the necessary nodes ($\sigma_1$ and one of the following:
$\sigma_3$ or $\sigma_6$ respectively $\sigma_2$ or $\sigma_3$).
\end{itemize}
\item[\bf $(\lambda_1+2\lambda_2)$--invariants:]
Since the homology class $A=\lambda_1+2\lambda_2$ contains only
one $\lambda_1$, all graphs $\mygraph$ in this homology class
contain exactly one of the following nodes: $\sigma_1$, $\sigma_2$
and $\sigma_3$. Therefore the following invariants are all zero:
\begin{list}{}{}
\item[$\Phi^{\lambda_1+2\lambda_2}(Z_1, Z_1Z_3Z_4, Z_1Z_4Z_5)=0$.]
\item[$\Phi^{\lambda_1+2\lambda_2}(Z_3, Z_1Z_3Z_4, Z_1Z_4Z_5)=0$.]
\item[$\Phi^{\lambda_1+2\lambda_2}(Z_1Z_3, Z_3Z_4, Z_1Z_4Z_5)=0$.]
\item[$\Phi^{\lambda_1+2\lambda_2}(Z_1Z_3, Z_1Z_3, Z_1Z_4Z_5)=0$.]
\end{list}
Only Gromov--Witten invariant in this class remains to be computed:
\begin{list}{}{}
\item[\bf $\Phi^{\lambda_1+2\lambda_2}(Z_4Z_5,Z_3Z_5, Z_1Z_3Z_4)=1$.]
Here we have to consider three graph types:

\noindent\begin{tabular}{cc}
$\bbull{\sigma_1} \mls \bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}$&
\\
$\bbull{\sigma_1} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}$&
\raisebox{0ex}[-5ex]{$\fork{\sigma_1}{\sigma_4}{\sigma_6}{\sigma_5}$}
\end{tabular}

\noindent The details of the computations are left to the reader.
\end{list}
\item[\bf $(\lambda_1+3\lambda_2)$--invariants are all zero:]
The virtual dimension of the corres\-pon\-ding moduli space is eight,
so we can set $\alpha_1=Z_1Z_3Z_4$ and $\alpha_2=Z_1Z_4Z_5$. A
graph $\mygraph$ that could give a non--zero integral on
$\cM_\mygraph$ had to contain $\sigma_1$ and $\sigma_3$, which is
impossible since the class $A=\lambda_1+3\lambda_2$ contains only
one $\lambda_1$.
\item[\bf $(2\lambda_1+\lambda_2)$--invariants:] Graph types for this class
have either on edge in the ``lower triangle'', or two side edges and one
on the ``upper triangle''.
\begin{list}{}{}
\item [\bf $\Phi^{2\lambda_1+\lambda_2}(Z_3Z_5, Z_1Z_3Z_4, Z_1Z_4Z_5)=0$.] Would need
two edges in the triangles.
\item[\bf $\Phi^{2\lambda_1+\lambda_2}(Z_2^2, Z_1Z_3Z_4, Z_1Z_4Z_5)=-2$.]
The only graph type we need to consider is:
\[ \gtype = \bbull{\sigma_1} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6}
\mls \bbull{\sigma_3} \]
By the usual computations we get $|\AA_{\gtype}|=1$ and
\begin{align*}
S_{\gtype}&= -2 (\we_{12}-\we_3-\we_5)(\we_3-\we_5)^2(\we_4-\we_5)(\we_4-\we_3)\\
T_{\gtype}&= -\left( (-\we_12+\we_3+\we_5)(\we_3-\we_5)^2(\we_4-\we_5)(\we_4-\we_3)\right)^{-1}.
\end{align*}
\item[\bf $\Phi^{2\lambda_1+\lambda_2}(Z_1Z_3, Z_1Z_3Z_4, Z_1Z_4Z_5)=1$.]
Here we need to consider the following two graph types for which we leave
the computations to the reader:
\[ \bbull{\sigma_1} \ml \bbull{\sigma_3} \qquad
\bbull{\sigma_1} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_3}
\]
\end{list}
\end{list}

\subsection{The quantum cohomology ring of $\P(\O_{\P^2}(2) \oplus 1)$}

For completeness, we will also compute the quantum cohomology ring of this manifold,
although it has already been known thanks to Givental's work \cite{giv97} where he uses
techniques from mirror symmetry to compute the quantum cohomology ring for Fano toric
varieties, obtaining the same formula postulated by Batyrev in \cite{bat93}.

Since the usual cohomology ring is given by
\[ H^*(\vt, \Q) = \Q[Z_1, Z_2, Z_3, Z_4, Z_5]/_{\left\langle
\displaystyle \begin{array}{l} Z_1-Z_2-2Z_5, Z_3-Z_4, Z_3-Z_5,\\
Z_1Z_2, Z_3Z_4Z_5 \end{array} \right\rangle,}\]
it suffices to calculate the quantum products $Z_1\star Z_2$ and $Z_3\star Z_3\star Z_3$
to find a representation of the quantum cohomology ring. Remember that given the
Gromov--Witten invariants, the quantum product satisfies the following determining
equalities:
\begin{equation}
\begin{split}
\langle (\alpha\star\beta)_\lambda\cap\gamma, [X]\rangle=&
\Phi^{X,\lambda}_{3,0}(\alpha,\beta,\gamma)\\
\alpha\star\beta=&\sum_{\lambda\in H^2(X,\Z)}(\alpha\star\beta)_\lambda q^\lambda,
\end{split}
\end{equation}
where $\alpha,\beta,\gamma\in H^*(X,\Q)$ are cohomology classes of the manifold $X$.
Thus if $\theta_1, \ldots, \theta_r$ is a basis of $H^*(X,\Q)$ and
$\vartheta_1, \ldots, \vartheta_r$ its
dual basis with respect to the cap product plus integration, we obtain
\[ (\alpha\star\beta)_\lambda = \sum_{i=1}^r \Phi^{X,\lambda}_{3,0}(\alpha,\beta, \theta_i)
\vartheta_i.\]

Now, for our particular example $X=X_\Sigma$, we will take the following basis with
its dual basis:
\[\begin{array}{l|c|c|c|c|c|c}
\mbox{basis}&1&Z_1&Z_3&Z_1Z_3&Z_3^2&Z_1Z_3^2\\\hline
\mbox{dual basis}&Z_1Z_3^2&Z_3^2&Z_1Z_3-2Z_3^2&Z_3&Z_1-2Z_3&1.
\end{array}
\]

So, for the first product $Z_1\star Z_2$ we obtain
\begin{eqnarray*}
Z_1\star Z_2&=&Z_1 \star Z_1 - 2Z_1\star Z_3\\
&=&(Z_1\star Z_1)_{\lambda_1} q^{\lambda_1}\\
&=& q^{\lambda_1}, \mbox{ since}\\
Z_1\star Z_1&=&Z_1^2 + q^{\lambda_1}\\
&=&\underbrace{Z_1Z_2}_{=0} + 2Z_1Z_3 + q^{\lambda_1}.
\end{eqnarray*}

For $Z_3^{\star3}$ we first calculate the quantum square product of $Z_3\star Z_3$:
\begin{eqnarray*}
Z_3\star Z_3&=&Z_3\cup Z_3 +
\underbrace{(Z_3\star Z_3)_{\lambda_2}}_{\Phi^{\lambda_2}(Z_3, Z_3, Z_3^2)(Z_1-2Z_3)}q^{\lambda_2}
+\underbrace{(Z_3\star Z_3)_{\lambda_1}}_{\Phi^{\lambda_1}(Z_3, Z_3, pt)=0}q^{\lambda_1}+\\
&&+ \underbrace{(Z_3\star Z_3)_{2\lambda_2}}_{\Phi^{2\lambda_2}(Z_3, Z_3, pt)=0}q^{2\lambda_2}\\
&=&Z_3^2 - (Z_1-2Z_3) q^{\lambda_2}.
\end{eqnarray*}

So we will also need the products $Z_1\star Z_3$ and $Z_3^2\star Z_3$. For the first product
notice that all Gromov--Witten invariants $\Phi^A(Z_1, Z_3, \alpha)$ are zero for $A\neq 0$.
So
\[ Z_1 \star Z_3 = Z_1\cup Z_3.\]
For the second product we obtain:
\begin{eqnarray*}
Z_3^2\star Z_3&=& Z_3^3 + (Z_3^2\star Z_3)_{\lambda_1}q^{\lambda_1} +
(Z_3^2\star Z_3)_{\lambda_2}q^{\lambda_2}+\\
&&+ (Z_3^2\star Z_3)_{2\lambda_2}q^{2\lambda_2} +(Z_3^2\star Z_3)_{3\lambda_2}q^{3\lambda_2}+
(Z_3^2\star Z_3)_{\lambda_1+\lambda_2}q^{\lambda_1+\lambda_2}\\
&=& -(Z_1Z_3-2Z_3^2)q^{\lambda_2}-2(Z_1-2Z_3)q^{2\lambda_2} + q^{\lambda_1+\lambda_2}
\end{eqnarray*}

\noindent Summing all up we thus get the following expression for $Z^{\star3}$:
\begin{eqnarray*}
Z_3\star Z_3\star Z_3&=&Z_3^2\star Z_3 + Z_1\star Z_3q^{\lambda_2}\\
&=& (-2Z_1Z_3+4Z_3^2)q^{\lambda_2}-4(Z_1-2Z_3)q^{2\lambda_2}+q^{\lambda_1+\lambda_2}\\
&=& (-2Z_1\star Z_3 +4 Z_3\star Z_3 + q^{\lambda_1})q^{\lambda_2}\\
&=&  Z_2\star Z_2 q^{\lambda_2}.
\end{eqnarray*}

Thus we obtain  the same result as Givental in \cite{giv97}, the quantum cohomology
ring defined by Batyrev in \cite{bat93}:

\begin{prop}
The quantum cohomology ring of the smooth toric Fano variety $\P_{\C\P^2}(\O(2)\oplus 1)$
is equal to
\begin{align*}
\lefteqn{QH^*(\P_{\C\P^2}(\O(2)\oplus 1), \C)=}\\ &=
\C[Z_1, Z_2,Z_3,Z_4, Z_5, q_1, q_2]/_{\left\langle
\begin{array}{l}Z_1-Z_2-2Z_5, Z_3-Z_4, Z_3-Z_5,\\ Z_1Z_2-q_1, Z_3Z_4Z_5-Z_2^2q_2\end{array}\right\rangle}\\
&= \C[Z_2, Z_3, q_1, q_2]/{\langle Z_2^2+2Z_2Z_3-q_1, Z_3^3-Z_2^2q_2\rangle}.
\end{align*}
\end{prop}

\subsection{On the quantum cohomology ring of the $3$--fold $\P(\O_{\P^2}(3) \oplus 1)$}

In this section, we will give the details of the computation of some
Gromov--Witten invariants of the (non--Fano) threefold
$\P(\O_{\P^2}(3) \oplus 1)$. As a corollary, we will show that --- in
the contrary to the previous example in the Fano case --- the
quantum cohomology ring of this variety defined by the Gromov--Witten
invariants of \cite{rt95,bf97,beh97,lt96,lt98a,sie96} does not
coincide with the ring defined formally in \cite{bat93}.

\subsubsection{Some Gromov--Witten invariants of $\P(\O_{\P^2}(3) \oplus 1)$}

In this section we will compute the genus--$0$ Gromov--Witten invariants
$\Phi^{p\lambda_2}(Z_3, Z_3, Z_3)$ and $\Phi^{p\lambda_2}(Z_2, Z_2, Z_2)$
for $p\leq 2$. Note first of all that since $Z_3^3=0$ in the cohomology
ring,
\[ \Phi^0(Z_3, Z_3, Z_3)=0.\]

\paragraph{The invariant $\Phi^{\lambda_2}(Z_3, Z_3, Z_3)$}

The $\lambda_2$--graphs have to live on the triangle in the moment
polytope spanned by $\sigma_4$, $\sigma_5$ and $\sigma_6$. Hence,
there are three $\lambda_2$--graphs:
\[ \gtype[1] = \bbull{\sigma_4} \ml[1] \bbull{\sigma_5} \qquad
\gtype[2] = \bbull{\sigma_4} \ml[1] \bbull{\sigma_6} \quad
\gtype[3] = \bbull{\sigma_5} \ml[1] \bbull{\sigma_6}
\]

First of all we note that none of the graphs has any non--trivial automorphisms.
Since the only edge of any of the graphs has multiplicity one we get
\[ |\AA_{\gtype[i]} | = 1. \]

We will now compute the $S_\Gamma$--terms for the three graphs. In fact,
there is only one such for each of the graphs, since the cohomology classes
are all equal to $Z_3$. By applying formula \ref{eq:sgammadef} we obtain:
\begin{align*}
S_{\gtype[1]}&= \left( \frac{1\cdot (\we_3-\we_5)}{(\we_4-\we_5)} + \frac{1\cdot (\we_3-\we_4)}{(\we_5-\we_4)}\right)^3\\
&=1^3=1.
\end{align*}
\begin{align*}
S_{\gtype[2]}&= \left( \frac{1\cdot (\we_3-\we_5)}{(\we_3-\we_5)} + 0 \right)^3 = 1\\
S_{\gtype[3]}&= \left( \frac{1\cdot (\we_3-\we_4)}{(\we_3-\we_4)} + 0 \right)^3 = 1.
\end{align*}

Let us now compute the $T_\Gamma$--terms. Remember, that $\lambda_e=\lambda_2$.
Hence we get:
\begin{align*}
T_{\gtype[1]}&= \left( \frac{(-1) \cdot 1^2}{1^2 \cdot (\we_4-\we_5)^2} \cdot
\frac{(-\we_1+\we_2+\we_4+2\we_5)(-\we_1+\we_2+2\we_4+\we_5)}{(\we_3-\we_5)(\we_3-\we_4)} \right)\cdot\\
&\phantom{=}\cdot \frac{1\cdot\left( \frac{1}{(\we_4-\we_5)} \right)^{-2}}{(\we_4-\we_5)}
\cdot \frac{1\cdot\left( \frac{1}{(\we_5-\we_4)} \right)^{-2}}{(\we_5-\we_4)}\\
&= \frac{(-\we_1+\we_2+\we_4+2\we_5)(-\we_1+\we_2+2\we_4+\we_5)}{(\we_3-\we_5)(\we_3-\we_4)}\\
T_{\gtype[2]}&= \left( \frac{(-1) \cdot 1^2}{1^2 \cdot (\we_3-\we_5)^2} \cdot
\frac{(-\we_1+\we_2+\we_3+2\we_5)(-\we_1+\we_2+2\we_3+\we_5)}{(\we_4-\we_5)(\we_4-\we_3)} \right)\cdot\\
&\phantom{=}\cdot \frac{1\cdot\left( \frac{1}{(\we_3-\we_5)} \right)^{-2}}{(\we_3-\we_5)}
\cdot \frac{1\cdot\left( \frac{1}{(\we_5-\we_3)} \right)^{-2}}{(\we_5-\we_3)}\\
&= \frac{(-\we_1+\we_2+\we_3+2\we_5)(-\we_1+\we_2+2\we_3+\we_5)}{(\we_4-\we_5)(\we_4-\we_3)}\\
T_{\gtype[3]}&= \left( \frac{(-1) \cdot 1^2}{1^2 \cdot (\we_3-\we_3)^2} \cdot
\frac{(-\we_1+\we_2+\we_3+2\we_4)(-\we_1+\we_2+2\we_3+\we_4)}{(\we_5-\we_4)(\we_5-\we_3)} \right)\cdot\\
&\phantom{=}\cdot \frac{1\cdot\left( \frac{1}{(\we_3-\we_4)} \right)^{-2}}{(\we_3-\we_4)}
\cdot \frac{1\cdot\left( \frac{1}{(\we_4-\we_3)} \right)^{-2}}{(\we_4-\we_3)}\\
&= \frac{(-\we_1+\we_2+\we_3+2\we_4)(-\we_1+\we_2+2\we_3+\we_4)}{(\we_5-\we_4)(\we_5-\we_3)}
\end{align*}

Therefore computing the sum $\sum_i (S_{\Gamma_i}\cdot T_{\Gamma_i})/|\AA_{\Gamma_i}|$ (for example
with the Maple package) yields
\[ \Phi^{\lambda_2}(Z_3, Z_3, Z_3) = 3.\]

\paragraph{The invariant $\Phi^{2\lambda_2}(Z_3, Z_3, Z_3)$}

As for the $\lambda_2$--invariant, this invariant has to live in the triangle with
corners $\sigma_4$, $\sigma_5$ and $\sigma_6$. In this case, we get twelve different graph
types:
\begin{xalignat*}{3}
\gtype[a]&=\bbull{\sigma_4} \mls[2] \bbull{\sigma_5}&
\gtype[aa]&=\bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_4}&
\gtype[ab]&=\bbull{\sigma_5} \mls \bbull{\sigma_4} \mls \bbull{\sigma_5}\\
\gtype[b]&=\bbull{\sigma_4} \mls[2] \bbull{\sigma_6}&
\gtype[ba]&=\bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_4}&
\gtype[bb]&=\bbull{\sigma_6} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6}\\
\gtype[c]&=\bbull{\sigma_5} \mls[2] \bbull{\sigma_6}&
\gtype[ca]&=\bbull{\sigma_5} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}&
\gtype[cb]&=\bbull{\sigma_6} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}\\
&&\gtype[d]&=\bbull{\sigma_4} \mls \bbull{\sigma_5} \mls \bbull{\sigma_6}\\
&&\gtype[e]&=\bbull{\sigma_4} \mls \bbull{\sigma_6} \mls \bbull{\sigma_5}\\
&&\gtype[f]&=\bbull{\sigma_5} \mls \bbull{\sigma_4} \mls \bbull{\sigma_6}
\end{xalignat*}

The order of the automorphism groups $\AA_{gtype[i]}$ is as follows:

\medskip

\begin{center}
\begin{tabular}{|l||c|c|c|}\hline
&&aa, ba, ca &\\
&a, b, c&ab, bb, cb&d, e, f\\\hline\hline
$|\Aut_{\gtype}|$&1&2&1\\\hline
$|\bigoplus \Z_{d_e}|$&2&1&1\\\hline\hline
$|\AA_{\gtype}|$&2&2&1\\\hline
\end{tabular}
\end{center}

\medskip

We will now compute the $S_\Gamma$--terms. Note that the images on the moment
polytope of the graphs $\gtype[a]$, $\gtype[aa]$ and $\gtype[ab]$ are all
the same, and hence there $S_{\gtype}$--terms coincide. The same applies to the
graphs $\gtype[b]$, $\gtype[ba]$ and $\gtype[bb]$, respectively
$\gtype[c]$, $\gtype[ca]$ and $\gtype[cb]$. Hence we obtain:
\[
S_{\gtype[a]}=S_{\gtype[aa]}=S_{\gtype[ab]}=
\left( \frac{2(\we_3-\we_5)}{(\we_4-\we_5)} + \frac{2(\we_3-\we_4)}{(\we_5-\we_4)}\right)^3
=2^3=8;
\]
\[
S_{\gtype[b]}=S_{\gtype[ba]}=S_{\gtype[bb]}=
\left(\frac{2(\we_3-\we_5)}{(\we_3-\we_5)} + 0 \right)^3 = 2^3 = 8;
\]
\[
S_{\gtype[c]}=S_{\gtype[ca]}=S_{\gtype[cb]}=
\left(\frac{2(\we_3-\we_4)}{(\we_3-\we_4)} + 0 \right)^3 = 2^3 = 8;
\]
\[
S_{\gtype[d]}=\left(\frac{(\we_3-\we_5)}{(\we_4-\we_5)} + \left(\frac{1}{(\we_5-\we_4)} +
\frac{1}{(\we_3-\we_4)}\right)(\we_3-\we_4) + 0 \right)^3= 2^3=8;
\]
\[
S_{\gtype[e]}=\left( \frac{(\we_3-\we_5)}{(\we_3-\we_5)} + 0 +
\frac{(\we_3-\we_4)}{(\we_3-\we_4)} \right)^3 = 2^3 = 8;
\]
\[
S_{\gtype[f]}=\left(\frac{(\we_3-\we_4)}{(\we_5-\we_4)} + \left(\frac{1}{(\we_3-\we_5)} +
\frac{1}{(\we_4-\we_5)}\right)(\we_3-\we_5) + 0 \right)^3= 2^3=8.
\]

All we are left with is computing the $T_{\gtype}$--terms and summing up everything. To
save some space and writing, we will write again $\we_{12}=\we_1-\we_2=-\we_{21}$. So
let us compute the $T_{\gtype}$--terms:
\begin{align*}
T_{\gtype[a]}&= \frac{(-1)^2 \cdot 2^4}{2^2 \cdot (\we_4-\we_5)^4} \cdot
\frac{(\we_{21}+\frac{5}{2}\we_5+\frac{1}{2}\we_4)\cdots
(\we_{21}+\frac{1}{2}\we_5+\frac{5}{2}\we_4)}{(\we_3-\we_5)(\we_3-\frac{1}{2}\we_4-\we_5)
(\we_3-\we_4)}\cdot\\
&\phantom{= }\cdot\frac{1\cdot\left(\frac{2}{(\we_4-\we_5)}\right)^{-2}}{\frac{(\we_4-\we_5)}{2}} \cdot
\frac{1\cdot\left(\frac{2}{(\we_5-\we_4)}\right)^{-2}}{\frac{(\we_5-\we_4)}{2}}\\
&= -
\frac{(\we_{21}+\frac{5}{2}\we_5+\frac{1}{2}\we_4)\cdots(\we_{21}+\frac{1}{2}\we_5+\frac{5}{2}\we_4)}{
(\we_4-\we_5)^2(\we_3-\we_5)(\we_3-\we_4)(\we_3-\frac{1}{2}\we_4-\frac{1}{2}\we_5)}.
\end{align*}
\begin{align*}
T_{\gtype[aa]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_4-\we_5)^2} \cdot
\frac{(\we_{21}+2\we_5+\we_4)(\we_{21}+\we_5+2\we_4)}{(\we_3-\we_5)(\we_3-\we_4)}
\right]^2 \cdot\\
&\phantom{= }\cdot\left[ \frac{1\cdot (\we_4-\we_5)^2}{(\we_4-\we_5)}\right]^2
 \cdot
\frac{(\we_{21}+3\we_4)(\we_3-\we_4)(\we_5-\we_4)\cdot
\left(\frac{2}{(\we_5-\we_4)}\right)^{-1}}{(\we_5-\we_4)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_4)(\we_{21}+2\we_5+\we_4)^2(\we_{21}+\we_5+2\we_4)^2}{(\we_4-\we_5)^2(\we_3-\we_5)^2(\we_3-\we_4)}.
\end{align*}
\begin{align*}
T_{\gtype[ab]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_4-\we_5)^2} \cdot
\frac{(\we_{21}+2\we_5+\we_4)(\we_{21}+\we_5+2\we_4)}{(\we_3-\we_5)(\we_3-\we_4)}
\right]^2 \cdot\\
&\phantom{=} \cdot\left[ \frac{1\cdot (\we_5-\we_4)^2}{(\we_5-\we_4)}\right]^2
\cdot
\frac{(\we_{21}+3\we_5)(\we_3-\we_5)(\we_4-\we_5)\cdot
\left(\frac{2}{(\we_4-\we_5)}\right)^{-1}}{(\we_4-\we_5)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_5)(\we_{21}+2\we_5+\we_4)^2(\we_{21}+\we_5+2\we_4)^2}{(\we_4-\we_5)^2(\we_3-\we_4)^2(\we_3-\we_5)}.
\end{align*}
\begin{align*}
T_{\gtype[b]}&= \frac{(-1)^2 \cdot 2^4}{2^2 \cdot (\we_3-\we_5)^4} \cdot
\frac{(\we_{21}+\frac{5}{2}\we_5+\frac{1}{2}\we_3)\cdots
(\we_{21}+\frac{1}{2}\we_5+\frac{5}{2}\we_3)}{(\we_4-\we_5)(\we_4-\frac{1}{2}\we_3-\we_5)
(\we_4-\we_3)}\cdot\\
&\phantom{= }\cdot\frac{1\cdot\left(\frac{2}{(\we_3-\we_5)}\right)^{-2}}{\frac{(\we_3-\we_5)}{2}} \cdot
\frac{1\cdot\left(\frac{2}{(\we_5-\we_3)}\right)^{-2}}{\frac{(\we_5-\we_3)}{2}}\\
&= -
\frac{(\we_{21}+\frac{5}{2}\we_5+\frac{1}{2}\we_3)\cdots(\we_{21}+\frac{1}{2}\we_5+\frac{5}{2}\we_3)}{
(\we_3-\we_5)^2(\we_4-\we_5)(\we_4-\we_3)(\we_4-\frac{1}{2}\we_3-\frac{1}{2}\we_5)}.
\end{align*}
\begin{align*}
T_{\gtype[ba]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_5-\we_3)^2} \cdot
\frac{(\we_{21}+2\we_3+\we_5)(\we_{21}+\we_3+2\we_5)}{(\we_4-\we_3)(\we_4-\we_5)}
\right]^2 \cdot\\
&\phantom{=} \cdot\left[ \frac{1\cdot (\we_3-\we_5)^2}{(\we_3-\we_5)}\right]^2
\cdot
\frac{(\we_{21}+3\we_3)(\we_4-\we_3)(\we_5-\we_3)\cdot
\left(\frac{2}{(\we_5-\we_3)}\right)^{-1}}{(\we_5-\we_3)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_3)(\we_{21}+2\we_3+\we_5)^2(\we_{21}+\we_3+2\we_5)^2}{(\we_5-\we_3)^2(\we_4-\we_5)^2(\we_4-\we_3)}.
\end{align*}
\begin{align*}
T_{\gtype[bb]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_5-\we_3)^2} \cdot
\frac{(\we_{21}+2\we_3+\we_5)(\we_{21}+\we_3+2\we_5)}{(\we_4-\we_3)(\we_4-\we_5)}
\right]^2 \cdot\\
&\phantom{=} \cdot\left[ \frac{1\cdot (\we_5-\we_3)^2}{(\we_5-\we_3)}\right]^2
\cdot
\frac{(\we_{21}+3\we_5)(\we_3-\we_5)(\we_4-\we_5)\cdot
\left(\frac{2}{(\we_3-\we_5)}\right)^{-1}}{(\we_3-\we_5)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_5)(\we_{21}+2\we_3+\we_5)^2(\we_{21}+\we_3+2\we_5)^2}{(\we_5-\we_3)^2(\we_4-\we_3)^2(\we_4-\we_5)}.
\end{align*}
\begin{align*}
T_{\gtype[c]}&= \frac{(-1)^2 \cdot 2^4}{2^2 \cdot (\we_3-\we_4)^4} \cdot
\frac{(\we_{21}+\frac{5}{2}\we_4+\frac{1}{2}\we_3)\cdots
(\we_{21}+\frac{1}{2}\we_4+\frac{5}{2}\we_3)}{(\we_5-\we_4)(\we_5-\frac{1}{2}\we_4-\we_3)
(\we_5-\we_3)}\cdot\\
&\phantom{=}\frac{1\cdot\left(\frac{2}{(\we_3-\we_4)}\right)^{-2}}{\frac{(\we_3-\we_4)}{2}} \cdot
\frac{1\cdot\left(\frac{2}{(\we_4-\we_3)}\right)^{-2}}{\frac{(\we_4-\we_3)}{2}}\\
&= -
\frac{(\we_{21}+\frac{5}{2}\we_4+\frac{1}{2}\we_3)\cdots(\we_{21}+\frac{1}{2}\we_4+\frac{5}{2}\we_3)}{
(\we_3-\we_4)^2(\we_5-\we_4)(\we_5-\we_3)(\we_5-\frac{1}{2}\we_4-\frac{1}{2}\we_3)}.
\end{align*}
\begin{align*}
T_{\gtype[ca]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_3-\we_4)^2} \cdot
\frac{(\we_{21}+2\we_4+\we_3)(\we_{21}+\we_4+2\we_3)}{(\we_5-\we_4)(\we_5-\we_3)}
\right]^2 \cdot\\
&\phantom{=} \cdot
\left[ \frac{1\cdot (\we_3-\we_4)^2}{(\we_3-\we_4)}\right]^2
\cdot
\frac{(\we_{21}+3\we_3)(\we_4-\we_3)(\we_5-\we_3)\cdot
\left(\frac{2}{(\we_4-\we_3)}\right)^{-1}}{(\we_4-\we_3)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_3)(\we_{21}+2\we_4+\we_3)^2(\we_{21}+\we_4+2\we_3)^2}{(\we_3-\we_4)^2(\we_5-\we_4)^2(\we_5-\we_3)}.
\end{align*}
\begin{align*}
T_{\gtype[cb]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_3-\we_4)^2} \cdot
\frac{(\we_{21}+2\we_4+\we_3)(\we_{21}+\we_4+2\we_3)}{(\we_5-\we_4)(\we_5-\we_3)}
\right]^2 \cdot\\
&\phantom{=} \cdot
\left[ \frac{1\cdot (\we_4-\we_3)^2}{(\we_4-\we_3)}\right]^2
\cdot
\frac{(\we_{21}+3\we_4)(\we_3-\we_4)(\we_5-\we_4)\cdot
\left(\frac{2}{(\we_3-\we_4)}\right)^{-1}}{(\we_3-\we_4)^2}\\
&=\frac{1}{2}
\frac{(\we_{21}+3\we_4)(\we_{21}+2\we_4+\we_3)^2(\we_{21}+\we_4+2\we_3)^2}{(\we_3-\we_4)^2(\we_5-\we_3)^2(\we_5-\we_4)}.
\end{align*}
\begin{align*}
T_{\gtype[d]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_4-\we_5)^2} \cdot
\frac{(\we_{21}+2\we_5+\we_4)(\we_{21}+\we_5+2\we_4)}{(\we_3-\we_5)(\we_3-\we_4)}
\right]\cdot\\
&\phantom{=}\cdot \left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_3-\we_4)^2} \cdot
\frac{(\we_{21}+2\we_4+\we_3)(\we_{21}+\we_4+2\we_3)}{(\we_5-\we_4)(\we_5-\we_3)}
\right] \cdot \\
&\phantom{=}\cdot
\frac{(\we_{21}+3\we_4)(\we_3-\we_4)(\we_5-\we_4)\cdot
\left(\frac{1}{(\we_3-\we_4)}+\frac{1}{(\we_5-\we_4)}\right)^{-1}}{(\we_3-\we_4)(\we_5-\we_4)}
\cdot \\
&\phantom{=} \cdot \frac{1\cdot
\left(\frac{1}{(\we_4-\we_5)}\right)^{-2}}{(\we_4-\we_5)} \cdot
\frac{1\cdot \left(
\frac{1}{(\we_4-\we_3)}\right)^{-1}}{(\we_4-\we_3)}\\
&=-
\frac{(\we_{21}+2\we_5+\we_4)(\we_{21}+\we_5+2\we_4)(\we_{21}+2\we_4+\we_3)
}{(\we_3-\we_5)^2
(\we_3-\we_4)(\we_5-\we_4)(\we_5+\we_3-2\we_4)}\cdot\\
&\mbox{\hspace{15em}} \cdot(\we_{21}+\we_4+2\we_3)(\we_{21}+3\we_4).
\end{align*}
\begin{align*}
T_{\gtype[e]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_3-\we_5)^2} \cdot
\frac{(\we_{21}+2\we_5+\we_3)(\we_{21}+\we_5+2\we_3)}{(\we_4-\we_5)(\we_4-\we_3)}
\right]\cdot\\
&\phantom{=}\cdot \left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_4-\we_3)^2} \cdot
\frac{(\we_{21}+2\we_3+\we_4)(\we_{21}+\we_3+2\we_4)}{(\we_5-\we_3)(\we_5-\we_4)}
\right] \cdot \\
&\phantom{=}\cdot
\frac{(\we_{21}+3\we_3)(\we_4-\we_3)(\we_5-\we_3)\cdot
\left(\frac{1}{(\we_4-\we_3)}+\frac{1}{(\we_5-\we_3)}\right)^{-1}}{(\we_4-\we_3)(\we_5-\we_3)}
\cdot \\
&\phantom{=}\cdot
\frac{1\cdot
\left(\frac{1}{(\we_3-\we_5)}\right)^{-2}}{(\we_3-\we_5)} \cdot
\frac{1\cdot \left(
\frac{1}{(\we_3-\we_4)}\right)^{-1}}{(\we_3-\we_4)}\\
&=-
\frac{(\we_{21}+2\we_5+\we_3)(\we_{21}+\we_5+2\we_3)(\we_{21}+2\we_3+\we_4)}{(\we_4-\we_5)^2
(\we_4-\we_3)(\we_5-\we_3)(\we_4+\we_5-2\we_3)}\cdot\\
&\mbox{\hspace{15em}} \cdot (\we_{21}+\we_3+2\we_4)(\we_{21}+3\we_3).
\end{align*}
\begin{align*}
T_{\gtype[f]}&=\left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_5-\we_4)^2} \cdot
\frac{(\we_{21}+2\we_4+\we_5)(\we_{21}+\we_4+2\we_5)}{(\we_3-\we_4)(\we_3-\we_5)}
\right]\cdot\\
&\phantom{=}\cdot \left[ \frac{(-1)\cdot 1^2}{1^2\cdot
(\we_3-\we_5)^2} \cdot
\frac{(\we_{21}+2\we_5+\we_3)(\we_{21}+\we_5+2\we_3)}{(\we_4-\we_5)(\we_4-\we_3)}
\right] \cdot \\
&\phantom{=}\cdot
\frac{(\we_{21}+3\we_5)(\we_3-\we_5)(\we_4-\we_5)\cdot
\left(\frac{1}{(\we_3-\we_5)}+\frac{1}{(\we_4-\we_5)}\right)^{-1}}{(\we_3-\we_5)(\we_4-\we_5)}
\cdot \\
&\phantom{=} \cdot
\frac{1\cdot
\left(\frac{1}{(\we_5-\we_4)}\right)^{-2}}{(\we_5-\we_4)} \cdot
\frac{1\cdot \left(
\frac{1}{(\we_5-\we_3)}\right)^{-1}}{(\we_5-\we_3)}\\
&=-
\frac{(\we_{21}+2\we_4+\we_5)(\we_{21}+\we_4+2\we_5)(\we_{21}+2\we_5+\we_3)}{(\we_3-\we_4)^2
(\we_3-\we_5)(\we_4-\we_5)(\we_4+\we_3-2\we_5)} \cdot\\
&\mbox{\hspace{15em}} \cdot(\we_{21}+\we_5+2\we_3)(\we_{21}+3\we_5) .
\end{align*}

Now, we just have to add up\footnote{For this fastidious exercise we have used the Maple package.}
all the twelve terms $S_\Gamma\cdot
T_\Gamma/|\AA_\Gamma|$ --- the result is:
\[ \Phi^{2\lambda_2}(Z_3, Z_3, Z_3) = -45.\]

\paragraph{The invariants $\Phi^{p\lambda_2}(Z_2, Z_2, Z_2)$}

In the cohomology ring, we have the relation $Z_2=Z_1-3Z_3$. Since
the Gromov--Witten invariant is linear and commutative in its arguments, we obtain
\begin{align*}
\Phi^{p\lambda_2}(Z_2, Z_2, Z_2) =& \Phi^{p\lambda_2}(Z_1,Z_1,Z_1)
- 9\Phi^{p\lambda_2}(Z_1,Z_1,Z_3) +\\
&+27\Phi^{p\lambda_2}(Z_1,Z_3,Z_3) -
27\Phi^{p\lambda_2}(Z_3,Z_3,Z_3).
\end{align*}
Now observe that the weight $\we_{\sigma(\gotv)}^{l_i}$ is zero for $l_i$
corresponding to $Z_1$ and $\gotv\in \{\gotv_4, \gotv_5,
\gotv_6\}$ in the triangle of the moment polytope where
$p\lambda_2$--graphs live. This yields
\[ \Phi^{p\lambda_2}(Z_2, Z_2, Z_2) = -27\Phi^{p\lambda_2}(Z_3, Z_3, Z_3),\]
and by the results from the previous subsections we get
\begin{align*}
\Phi^0(Z_2,Z_2,Z_2)&=Z_2^3=9Z_1Z_3^2=9\\
\Phi^{\lambda_2}(Z_2,Z_2,Z_2)&=-27\cdot 3 = -81\\
\Phi^{2\lambda_2}(Z_2,Z_2,Z_2)&=-27\cdot (-45)=1215.
\end{align*}

\subsubsection{Conclusions for the quantum cohomology ring of $\P(\O_{\P^2}(3) \oplus 1)$}

In \cite[Definition 5.1]{bat93}, Batyrev defines
the quantum cohomology ring of a symplectic toric manifold
$X_\Sigma$ with symplectic form\footnote{Remember that two--forms
on $X_\Sigma$ can be represented by piecewise linear functions on the fan $\Sigma$}
$\varphi$ by (using our notation)
\[ QH^*_\varphi(X_\Sigma, \C) := \C[Z_1, \ldots, Z_n]
\left/_{\textstyle ({\rm Lin}(\Sigma) + Q_\varphi(\Sigma)),}\right.\]
where the ideal $Q_\varphi(\Sigma)$ is generated by the monomials
$B_\varphi(P)$:
\[ B_\varphi(P) = Z_{i_1} \cdots Z_{i_k} - E_\varphi(P) Z_{j_1}^{c_1}\cdots Z_{j_l}^{c_l},\]
where $P \in {\mathfrak P}$ runs over all primitive collections of
$\Sigma$, and
\begin{enumerate}
\item for a primitive collection $P=\{\gotv_{i_1}, \ldots
\gotv_{i_k}\}$, let $\gotv_P:=\gotv_{i_1}+\cdots+\gotv_{i_k}$ be
the sum of these vectors, $\sigma_P$ be the minimal cone that
contains the vector $\gotv_P$, $\gotv_{j_1}, \ldots, \gotv_{j_l}$
be the generators of $\sigma_P$, and $c_1, \ldots, c_l$ be the
integral coefficients of $\gotv_P$ with respect to these
generators:
\[ \gotv_P = c_1\gotv_{j_1} + \cdots + c_l\gotv_{j_l};\]
\item $E_\varphi(P):=
\exp(c_1\varphi(\gotv_{j_1})+\cdots+c_l\varphi(\gotv_{j_l}) -
\varphi(\gotv_{i_1})-\cdots-\varphi(\gotv_{i_l})).$
\end{enumerate}

Batyrev shows in \cite{bat93} that the structure constants of
this quantum cohomology ring are some intersection products on the
moduli space of holomorphic mappings $f: \C\P^1 \longrightarrow
X_\Sigma$. In fact, these intersection products are almost the
Gromov--Witten invariants (that where not properly defined at the
time): the only difference is that Batyrev does not compactify his
moduli space of maps.

It is therefore very interesting to know whether the quantum
cohomology ring obtained from Gromov--Witten invariants as
structure constants coincided with Batyrev's ring, \ie. to know
whether the boundary components of the moduli space have any
effect on the quantum product.

Givental has given in \cite{giv97}
an affirmative answer for Fano toric manifolds: there the two
rings are the same. For non--Fano manifolds, nothing was known so
far to the author's knowledge.

Let us suppose the quantum cohomology ring of $X_\Sigma$ was as
proposed by Batyrev\footnote{Instead of the $\exp(\ldots)$--terms we
will introduce formal variables. This is equivalent to Batyrev's notion of
the ring, since his formulas do not depend on the K\"ahler form $\varphi$.}:
\[
QH^*(X_\Sigma)=\C[Z_1, \ldots,
Z_5, q^{\lambda_1}, q^{\lambda_2}]\left/_{\left\langle
\begin{array}{l} Z_3-Z_5, Z_4-Z_5,
Z_1-Z_2-3Z_5,\\
Z_1Z_2-q^{\lambda_1},
Z_3Z_4Z_5-Z_2^3q^{\lambda_2}\end{array}\right\rangle.}\right.
\]
The relation
$Z_3Z_4Z_5-Z_2^3q^{\lambda_2}$ implies on the level of
Gromov--Witten invariants that
\[ \forall \alpha\in H^*(X_\Sigma),
A\in H_2(X_\Sigma) : \, \Phi^{A+\lambda_2}(Z_3, Z_4, Z_5, \alpha)
= \Phi^A(Z_2, Z_2, Z_2, \alpha).
\]
Taking $A=p\lambda_2$ to be a
multiple of the class $\lambda_2$, and $\alpha=P.D.(point)$ to be
the Poincar\'e dual of a point, this boils down to
$\Phi^{(p+1)\lambda_2}(Z_3, Z_3, Z_3) =
\Phi^{p\lambda_2}(Z_2,Z_2,Z_2)$ for all $p\in {\mathbb N}$, which is obviously not satisfied
for $p$ as small as $p=0$ and $p=1$.

\begin{coro}
Batyrev's definition as in \cite{bat93} of the
quantum cohomology ring of a symplectic toric manifold does not
coincide with the one obtained from Gromov--Witten invariants as
defined in \cite{bf97} for the manifold $X_\Sigma=P_{{\C\P}^2}({\O}(3) \oplus 1)$.
\end{coro}

\begin{rema}
The example given above is not the simplest one can give for a manifold
where the quantum cohomology ring defined by Gromov--Witten invariants does
not coincide with Batyrev's ring. Consider the class of Fano surfaces
$F_k:=\P_{\C\P^1}(O(k)\oplus 1)$. All Fano surfaces are K\"ahler manifolds, and
for two different parameters $k_1, k_2 \in \Z$ $F_{k_1}$ and $F_{k_2}$ are in
the same deformation class of symplectic manifolds if and only if $k_1 \equiv
k_2 \mod 2$ (otherwise they are topologically different). Hence, via the
diffeomorphism identifying $F_{k_1}$ and $F_{k_2}$, the Gromov--Witten invariants
are the same, and so is the quantum cohomology ring.

Batyrev's rings, however, are not identified by this diffeomorphism.
\end{rema}