<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_cotangent__Sheaf_lp__Z__Z_cm__Projective__Variety_rp.html">next</a> | <a href="_cotangent__Sheaf.html">previous</a> | <a href="_cotangent__Sheaf_lp__Z__Z_cm__Projective__Variety_rp.html">forward</a> | <a href="_cotangent__Sheaf.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>cotangentSheaf X</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_cotangent__Sheaf.html" title="cotangent sheaf of a projective variety">cotangentSheaf</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>X</tt>, <span>a <a href="___Projective__Variety.html">projective variety</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span>, the cotangent sheaf of the projective variety</span></li> </ul> </div> </li> <li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Minimize => </tt><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <span>default value true</span>, whether to apply <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> to the result before returning it</span></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>This function computes the cotangent sheaf of the projective variety <tt>X.</tt><p/> As an example we verify Gauss-Bonnet's theorem on a plane quartic curve:<table class="examples"><tr><td><pre>i1 : X=Proj(QQ[x,y,z]/(x^4+y^4+z^4)) o1 = X o1 : ProjectiveVariety</pre> </td></tr> <tr><td><pre>i2 : genus(X) o2 = 3</pre> </td></tr> <tr><td><pre>i3 : omega = cotangentSheaf(X) 1 o3 = OO (1) X o3 : coherent sheaf on X, free</pre> </td></tr> <tr><td><pre>i4 : degree omega o4 = 4</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_tangent__Sheaf.html" title="tangent sheaf of a projective variety">tangentSheaf</a> -- tangent sheaf of a projective variety</span></li> <li><span><a href="___Projective__Variety.html" title="the class of all projective varieties">ProjectiveVariety</a> -- the class of all projective varieties</span></li> </ul> </div> </div> </body> </html>