most recent 30 from http://mathoverflow.net 2013-05-25T15:43:58Z http://mathoverflow.net/feeds/question/80 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80/what-is-interesting-useful-about-castelnuovo-mumford-regularity David Zureick-Brown 2009-10-03T09:52:05Z 2009-10-06T01:58:01Z

<p>What is interesting/useful about Castelnuovo-Mumford regularity?</p>

http://mathoverflow.net/questions/80/what-is-interesting-useful-about-castelnuovo-mumford-regularity/96#96 Ilya Nikokoshev 2009-10-04T21:41:04Z 2009-10-04T22:01:59Z

<p>Here's an example paper: <strong><a href="http://arxiv.org/abs/0905.2212" rel="nofollow">0905.2212</a></strong></p> <p>It uses some bound on Castelnuovo–Mumford regularity to prove that cohomology of smooth complex projective variety can be computed in parallel polynomial time.</p>

http://mathoverflow.net/questions/80/what-is-interesting-useful-about-castelnuovo-mumford-regularity/127#127 Daniel Erman 2009-10-06T01:58:01Z 2009-10-06T01:58:01Z

<p>Here's how I think about Castelnuovo-Mumford regularity. It's an invariant of an ideal (or module or sheaf) which provides a measure of how complicated that ideal (or module or sheaf) is. Thsis invariant is related to free resolutions, and thus it measures complexity from that perspective.</p> <p>Why is it interesting? One answer is that it can be used to provide an effective bound for two famous theorems. The first theorem I have in mind is that the Hilbert function of a graded ideal (or a finitely generated graded module) over the polynomial ring eventually agrees with the Hilbert polynomial of that ideal (or module). The second theorem I have in mind is Serre vanishing, which says that, given a coherent sheaf F on PP^n, there exists d such that H^i(PP^n, F(e))=0 for all i>0 and all e>d. These two theorems are related: if M is a graded module of depth > 0, and F is the associated sheaf of M, then the Hilbert function of M in degree e equals H^0(PP^n,F(e)).</p> <p>An example where Castelnuovo-Mumford is particularly useful comes from the construction of the Hilbert scheme (I have heard that this is related to Mumford's original use, though I have no reference.) The basic point is that you can parametrize the set of ideals with a given Hilbert function by considering subloci of certain Grassmanians satisfying determinantal criteria, whereas it's less clear (at least to me) how to parametrize ideals with a given Hilbert polynomial.</p> <p>Another great example where Castelnuovo-Mumford is useful is presented in Eisenbud "The Geometry of Syzygies" chapter 4, where he solves the interpolation problem for points in affine space.</p>