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.

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 $C_i \equiv C_j$?

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.