Components of an exceptional divisor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:26:11Z http://mathoverflow.net/feeds/question/73737 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73737/components-of-an-exceptional-divisor Components of an exceptional divisor unknown (google) 2011-08-26T07:12:06Z 2013-02-21T19:29:28Z <p>Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder: </p> <p><em>What is the number of irreducible components of $F$?</em></p> <p>Note that this number depends strongly on the scheme structure on $Z$. For example, when $Z$ is a line in $\mathbb{P}^2$ with an embedded point, $F$ has two components, whereas the blow-up of a line has only one. So is it true in general that the number of components of $F$ is at least the number of associated primes of $Z$? (I am mostly interested in a lower bound for this number.)</p> http://mathoverflow.net/questions/73737/components-of-an-exceptional-divisor/76687#76687 Answer by Karl Schwede for Components of an exceptional divisor Karl Schwede 2011-09-28T20:22:19Z 2011-09-29T00:36:36Z <p>I had a minor thought. Did you ever look the paper <em>Multiplicity of the special fiber of blowups</em> by Corso,Polini, Vasconcelos. </p> <p>In particular, they give an upper bound on the number (I realize that you are interested in lower bounds, but maybe some of the ideas are related / or could be useful) by in particular, bounding the multiplicity at the origin of the Rees algebra fiber ring (in other words, if you compute the blow-up by computing Proj $R[It]$, then mod out by an ideal from $R$, you get some graded ring corresponding to the fiber over the ideal you modded. Then you can study the blow-up by studying properties of the ring and I think the multiplicity they study should give you an upper bound on the number of components).</p> <p>Of course, the number of minimal associated primes gives you some bound on the components of the pre-image of $Z$.</p> http://mathoverflow.net/questions/73737/components-of-an-exceptional-divisor/122567#122567 Answer by Dmitry Kerner for Components of an exceptional divisor Dmitry Kerner 2013-02-21T19:29:28Z 2013-02-21T19:29:28Z <p>I'd say the number can be arbitrary large if $X$ is not assumed smooth along $Z$. Take the simplest example: hypersurface singularity ${f_p+f_{p+1}=0}\subset\Bbb C^n$, where $f_p$ is totally reducible and $f_{p+1}$ is generic enough. So that the singularity is isolated. Blowup the origin. The exceptional divisor (=the projectivization of the tangent cone) is the hyperplane arrangement, ${f_p=0}\subset\Bbb P^{n-1}$. </p> <p>(Of course, you can compactify the hypersurface, if you insist on projective varieties)</p>