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<head><title>genus(ProjectiveVariety) -- arithmetic genus</title>
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<div><h1>genus(ProjectiveVariety) -- arithmetic genus</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>genus V</tt></div>
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<li><span>Function: <a href="_genus.html" title="arithmetic genus">genus</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>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div>Computes the arithmetic genus of the projective scheme <tt>V</tt>A nodal plane cubic curve has arithmetic genus 1:<table class="examples"><tr><td><pre>i1 : V = Proj(QQ[a,b,c]/ideal(b^2*c-a^2*(a+c)))
o1 = V
o1 : ProjectiveVariety</pre>
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<tr><td><pre>i2 : genus V
o2 = 1</pre>
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The Fano model of a Reye type Enriques surface in projective fivespace:<table class="examples"><tr><td><pre>i3 : R = ZZ/101[x_0..x_5];</pre>
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<tr><td><pre>i4 : M = random(R^4, R^{4:-1});
4 4
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : I = minors(3, M+transpose(M));
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : V = Proj(R/I);</pre>
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<tr><td><pre>i7 : genus V
o7 = 0</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_genera.html" title="list of the successive linear sectional arithmetic genera">genera</a> -- list of the successive linear sectional arithmetic genera</span></li>
<li><span><a href="_euler.html" title="Euler characteristic">euler</a> -- Euler characteristic</span></li>
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