Hi,


I'm trying to understand the group of cycles (modulo numerical equivalence) contracted by a flopping contraction $f$.


More precisely, I'm in the setup of Definition 2.12 of [this paper by Yukinobu Toda](http://arxiv.org/abs/0909.5129). 


Let $f: X \to Y$ be a flopping contraction: $X$ is a smooth and projective CY3, f is birational, $Y$ is Gorenstein, $f$ is isomorphic in codimension one, $dim\_\mathbb{R} N^1(X/Y)\_\mathbb{R}=1$. 


Where $N^1(X/Y)$ is the group of divisors of $X$ modulo numerical equivalence over $Y$ (viz. $D\_1 \equiv D\_2$ iff $D\_1.C=D\_2.C$ for all curves $C$ contracted by $f$).


(a side question is: what's the correct way to define "isomorphic in codimension d"?)


Denote, $N\_1(X/Y)$ the group of 1-cycles contracted by $f$, modulo numerical equivalence.



> 
> What is $N\_1(X/Y)$? (without tonsuring with Q or R)
> 
> 
> 


In the paper cited above, it is written that the exceptional locus of $f$ is a tree of projective lines $C\_1 \cup \ldots \cup C\_m$



> 
> Is $C\_i \equiv C\_j$?
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> 
> 


In the end I'm really hoping that $N\_1(X/Y) = \mathbb{Z}$. If this is not the case, then I'm also interested in what happens after tensoring with $\mathbb{Q}$.


Thanks.



What is the Exceptional Locus of a flopping contraction between threefolds?
1) $f$ is *isomorphic in codimension $d$* if it is an isomorphism near any codimension $d$ point in either $X$ or $Y$. Equivalently, there exists closed subsets $Z\subseteq X$ and $W\subseteq Y$ such that ${\rm codim}\_XZ\geq d+1$, ${\rm codim}\_YW\geq d+1$, and $f:X\setminus Z\overset{\simeq}{\longrightarrow} Y\setminus W$ is an isomorphism. 


2) By the Theorem of the Base of Néron–Severi, if $f$ is proper of finite type, then $N\_1(X/Y)\_{\mathbb Q}$ and $N^1(X/Y)\_{\mathbb Q}$ are finite-dimensional vector spaces of the same dimension. This is actually more than you need, because even without the finite type assumption it is true that the intersection pairing $N\_1(X/Y)\_{\mathbb Q}\times N^1(X/Y)\_{\mathbb Q}\to {\mathbb Q}$ is non-degenerate.


