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<head><title>monomialCurveIdeal -- make the ideal of a monomial curve</title>
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<div><h1>monomialCurveIdeal -- make the ideal of a monomial curve</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>I = monomialCurveIdeal(R,a)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span>, </span></li>
<li><span><tt>a</tt>, a list of integers to be used as exponents in the parametrization of a rational curve</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, </span></li>
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<div class="single"><h2>Description</h2>
<div><tt>monomialCurveIdeal(R,a)</tt> yields the defining ideal of the projective curve given parametrically on an affine piece by t |---> (t^a1, ..., t^an).<p/>
The ideal is defined in the polynomial ring R, which must have at least n+1 variables, preferably all of equal degree. The first n+1 variables in the ring are usedFor example, the following defines a plane quintic curve of genus 6.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : monomialCurveIdeal(R,{3,5})
5 2 3
o2 = ideal(b - a c )
o2 : Ideal of R</pre>
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Here is a genus 2 curve with one singular point.<table class="examples"><tr><td><pre>i3 : monomialCurveIdeal(R,{3,4,5})
2 2 2 3
o3 = ideal (c - b*d, b c - a*d , b - a*c*d)
o3 : Ideal of R</pre>
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Here is one with two singular points, genus 7.<table class="examples"><tr><td><pre>i4 : monomialCurveIdeal(R,{6,7,8,9,11})
2 2 2 2
o4 = ideal (e - c*f, d*e - b*f, d - c*e, c*d - b*e, c - b*d, b*c*e - a*f ,
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2 2 3
b d - a*e*f, b c - a*d*f, b - a*c*f)
o4 : Ideal of R</pre>
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Finally, here is the smooth rational quartic in P^3.<table class="examples"><tr><td><pre>i5 : monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o5 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o5 : Ideal of R</pre>
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