What is the Exceptional Locus of a flopping contraction between threefolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:59:14Z http://mathoverflow.net/feeds/question/44154 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44154/what-is-the-exceptional-locus-of-a-flopping-contraction-between-threefolds What is the Exceptional Locus of a flopping contraction between threefolds? babubba 2010-10-29T16:49:53Z 2010-10-29T21:28:39Z <p>Hi,</p> <p>I'm trying to understand the group of cycles (modulo numerical equivalence) contracted by a flopping contraction $f$.</p> <p>More precisely, I'm in the setup of Definition 2.12 of <a href="http://arxiv.org/abs/0909.5129" rel="nofollow">this paper by Yukinobu Toda</a>. </p> <p>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, $dim_\mathbb{R} N^1(X/Y)_\mathbb{R}=1$. </p> <p>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 $f$).</p> <p>(a side question is: what's the correct way to define "isomorphic in codimension d"?)</p> <p>Denote, $N_1(X/Y)$ the group of 1-cycles contracted by $f$, modulo numerical equivalence.</p> <blockquote> <p>What is $N_1(X/Y)$? (without tonsuring with Q or R)</p> </blockquote> <p>In the paper cited above, it is written that the exceptional locus of $f$ is a tree of projective lines $C_1 \cup \ldots \cup C_m$</p> <blockquote> <p>Is $C_i \equiv C_j$?</p> </blockquote> <p>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 $\mathbb{Q}$.</p> <p>Thanks.</p> http://mathoverflow.net/questions/44154/what-is-the-exceptional-locus-of-a-flopping-contraction-between-threefolds/44161#44161 Answer by Sándor Kovács for What is the Exceptional Locus of a flopping contraction between threefolds? Sándor Kovács 2010-10-29T17:51:01Z 2010-10-29T20:14:04Z <p>1) $f$ is <em>isomorphic in codimension $d$</em> 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. </p> <p>2) By the Theorem of the Base of Néron–Severi, if $f$ is proper of finite type, then <code>$N_1(X/Y)_{\mathbb Q}$</code> 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 <code>$N_1(X/Y)_{\mathbb Q}\times N^1(X/Y)_{\mathbb Q}\to {\mathbb Q}$</code> is non-degenerate.</p>