Implement intersection products - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:05:07Z http://mathoverflow.net/feeds/question/18543 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18543/implement-intersection-products Implement intersection products Dung Nguyen 2010-03-18T03:05:01Z 2010-03-18T14:48:34Z <p>I am doing a counting problem, and it comes to compute intersection products ( Chow ring ) on some varieties. Is there any computer algebra that deals with this?</p> http://mathoverflow.net/questions/18543/implement-intersection-products/18560#18560 Answer by David Holmes for Implement intersection products David Holmes 2010-03-18T08:14:27Z 2010-03-18T08:14:27Z <p>It would help to know the context of your work better - what category are you working in?</p> <p>I currently work computing arithmetic (Arakelov) intersection pairings on curves, for which I use magma, doing the actual calculations using either resultants or Grobner bases (at least at finite primes); all you are calculating are the lengths of modules, which is fairly standard for commutative algebra software I assume.</p> http://mathoverflow.net/questions/18543/implement-intersection-products/18576#18576 Answer by Jan-Magnus Økland for Implement intersection products Jan-Magnus Økland 2010-03-18T12:23:13Z 2010-03-18T12:23:13Z <p>Schubert2 in <a href="http://www.math.uiuc.edu/Macaulay2/" rel="nofollow">Macaulay2</a> and the original maple package <a href="http://folk.uib.no/nmasr/schubert/" rel="nofollow">schubert</a> let you build enough of the chow rings of parameter spaces to tackle some enumerative geometry problems like those of "Kalkül der abzählenden Geometrie" by H. Schubert (1879):</p> <h3>Count the number of space conics intersecting 8 given lines</h3> <blockquote> <p>> with(schubert):</p> <p>> grass(3,4,d,all):</p> <p>> Proj(f,dual(symm(2,Qd)),e):</p> <p>> integral(Gd,lowerstar(f,(2*d1+e)^8));</p> <pre><code> 92 </code></pre> </blockquote> <p>(Example 3.2.22 of "Intersection Theory" by W. Fulton (1984))</p> http://mathoverflow.net/questions/18543/implement-intersection-products/18592#18592 Answer by David Speyer for Implement intersection products David Speyer 2010-03-18T14:48:34Z 2010-03-18T14:48:34Z <p>For Grassmannians, see Buch's <a href="http://www.math.rutgers.edu/~asbuch/lrcalc/" rel="nofollow">Littlewood-Richardson calculator</a>. Buch <a href="http://www.math.rutgers.edu/~asbuch/qcalc/" rel="nofollow">quantum calculator</a> can compute quantum cohomology (and hence ordinary) on grassmannians, isotropic grassmannians and symplectic isotoropic grassmannians. <a href="http://orms.mfo.de/project?terms=polynomial&amp;id=307" rel="nofollow">Symmetrica</a> can compute Schubert polynomials, and hence should be able to be coerced into computing in the cohomology ring of the flag manifold, although I don't know how easy or hard that is.</p>