When is a scheme a zero-set of a section of a vector bundle? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:38:26Z http://mathoverflow.net/feeds/question/1614 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle When is a scheme a zero-set of a section of a vector bundle? Timo Schürg 2009-10-21T09:34:35Z 2010-09-30T12:47:23Z <p>Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M? To put it in a diagram: When is X the fiber product of M -> E &lt;- M , where one arrow is the zero section and the other arrow is the section I'm looking for. </p> <p>If this is not possible, can X be written as a degeneracy locus?</p> http://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle/1615#1615 Answer by David Lehavi for When is a scheme a zero-set of a section of a vector bundle? David Lehavi 2009-10-21T09:44:44Z 2009-10-21T09:44:44Z <p>As for the first question, the class of X has to be the product of the Chern roots of the bundle, so in the Chow ring, it is the class of a complete intersection.</p> <p>As for the second question, you would have to find classes that will solve the class of X in the Thom-Porteus formula, see Fulton's intersection theory 14.4</p> http://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle/2060#2060 Answer by Joel Fine for When is a scheme a zero-set of a section of a vector bundle? Joel Fine 2009-10-23T09:11:25Z 2009-10-23T09:11:25Z <p>At least when the subvariety has codimension 2, this is known as "the Serre construction". There's a nice description of the case of points in a surface given in "Lectures on linear series" by Lazarsfeld. I'm sure there are many other excellent references too, but that's the first that comes to mind.</p> http://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle/2086#2086 Answer by David Speyer for When is a scheme a zero-set of a section of a vector bundle? David Speyer 2009-10-23T13:15:56Z 2009-10-23T13:15:56Z <p>A necessary condition is that it be a locally complete intersection, since locally this is the same as asking that your scheme be the zero set of codimension many equations.</p> http://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle/40613#40613 Answer by Angelo for When is a scheme a zero-set of a section of a vector bundle? Angelo 2010-09-30T12:47:23Z 2010-09-30T12:47:23Z <p>Are you assuming that the rank of \$E\$ equals the codimension of the subscheme? You don't say so explicitly. If not, the answer is that every closed subscheme is a zero section, since it is the intersection of finitely many hypersurfaces.</p>