<?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>euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_euler_lp__Ring_rp.html">next</a> | <a href="_euler_lp__Projective__Hilbert__Polynomial_rp.html">previous</a> | <a href="_euler_lp__Ring_rp.html">forward</a> | <a href="_euler_lp__Projective__Hilbert__Polynomial_rp.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>euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) 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>euler V</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_euler.html" title="Euler characteristic">euler</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>V</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>an <a href="___Z__Z.html">integer</a></span>, the topological Euler characteristics of the variety V</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The command computes the topological Euler characteristic of the (smooth) projective variety V as an alternated sum of its Hodge numbers. The Hodge numbers can be computed directly using the command <a href="_hh.html" title="Hodge numbers of a smooth projective variety">hh</a>.<p/> A smooth plane quartic curve has genus 3 and topological Euler characteristic -4:<table class="examples"><tr><td><pre>i1 : Quartic = Proj(QQ[x_0..x_2]/ideal(x_0^4+x_1^4+x_2^4)) o1 = Quartic o1 : ProjectiveVariety</pre> </td></tr> <tr><td><pre>i2 : euler(Quartic) o2 = -4</pre> </td></tr> </table> <p/> The topological Euler characteristic of a smooth quintic hypersurface in projective fourspace is -200:<table class="examples"><tr><td><pre>i3 : Quintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-101*x_0*x_1*x_2*x_3*x_4)) o3 = Quintic o3 : ProjectiveVariety</pre> </td></tr> <tr><td><pre>i4 : euler(Quintic) o4 = -200</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div>No test is made to see if the projective variety is smooth</div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="___Proj_lp__Ring_rp.html" title="make a projective variety">Proj</a> -- make a projective variety</span></li> <li><span><a href="_genus.html" title="arithmetic genus">genus</a> -- arithmetic genus</span></li> <li><span><a href="_hh.html" title="Hodge numbers of a smooth projective variety">hh</a> -- Hodge numbers of a smooth projective variety</span></li> </ul> </div> </div> </body> </html>