User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:55:45Z http://mathoverflow.net/feeds/user/14199 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/79195/generalized-linear-systems Generalized linear systems unknown (google) 2011-10-26T21:30:22Z 2011-10-28T13:15:06Z <p>Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly equivalent to $[Z]$. When $Z$ is an effective divisor, this set is a linear system, and so has the structure of a projective space.</p> <p><em>Is there any reasonable geometric structure on $L(Z)$ in general?</em> </p> <p>If so, has this problem been studied before? In particular, can one give estimates for the dimension of $L(Z)$? </p> http://mathoverflow.net/questions/73711/the-concept-of-duality/73763#73763 Answer by unknown (google) for The concept of Duality unknown (google) 2011-08-26T11:07:24Z 2011-08-26T11:07:24Z <p>Serre duality</p> <p>Grothendieck duality</p> <p>Verdier duality</p> http://mathoverflow.net/questions/62137/ample-line-bundle-and-duality Ample line bundle and Duality unknown (google) 2011-04-18T15:35:10Z 2011-04-19T03:03:48Z <p>Let $X$ be a projective variety and let $L$ be a line bundle on $X$. Suppose for all locally free sheaves $M$ on $X$, $H^i(X,{L^*}^{\otimes r} \otimes M)=0$ for $i&lt;\dim X$ and $r$ sufficiently big. </p> <p>Does it follow that $L$ is an ample line bundle? Here $L^*$ denotes the dual of $L$.</p> <p>This is of course clear if $X$ is smooth using Serre duality, but how is it in general?</p> <p>After reading Laurent Moret-Bailly and Karl Schwede's comments, below I changed the condition '$M$ coherent' to '$M$ locally free'.</p> http://mathoverflow.net/questions/62137/ample-line-bundle-and-duality Comment by 2011-04-18T16:27:01Z 2011-04-18T16:27:01Z Yes, thanks, I edited.