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<head><title>distinguished -- compute the distinguished subvarieties of a scheme</title>
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<div><h1>distinguished -- compute the distinguished subvarieties of a scheme</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>distinguished I</tt><br/><tt>distinguished(I,f)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, optional argument, if given it should be a non-zero divisor in the ideal <tt>I</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_symmetric__Kernel_lp..._cm_sp__Variable_sp_eq_gt_sp..._rp.html">Variable => ...</a>, -- Choose name for variables in the created ring</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Let <i>I⊂R</i> be an ideal in a ring <i>R</i>, the image of a free <i>R</i>-module <i>F</i>. Let <i>ReesI</i> be the Rees algebra of <i>I</i>. Certain of the minimal primes of <i>I</i> are distinguished from the point of view of intersection theory: These are the ones that correspond to primes <i>P⊂ReesI</i> minimal among those containing <i>I*ReesI</i>---in other words, the isolated components of the support of the normal cone of <i>I</i>. The prime <i>p</i> corresponding to <i>P</i> is simply the kernel of the the induced map <i>R →ReesI/P</i>.</p>
<p>Each of these primes comes equipped with a multiplicity, which may be computed as the ratio <i>degree(ReesI/P)/degree(R/p)</i>.</p>
<p>For these matters and their significance, see section 6.1 of the book “Intersection Theory,” by William Fulton, and the references there, along with the paper</p>
<p>“A geometric effective Nullstellensatz.” Invent. Math. 137 (1999), no. 2, 427--448 by Ein and Lazarsfeld.</p>
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<table class="examples"><tr><td><pre>i1 : T = ZZ/101[c,d];</pre>
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<tr><td><pre>i2 : D = 4;</pre>
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<tr><td><pre>i3 : P = product(D, i -> random(1,T))
4 3 2 2 3 4
o3 = - 34c + 41c d + 30c d + 7c*d - 36d
o3 : T</pre>
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<tr><td><pre>i4 : R = ZZ/101[a,b,c,d]
o4 = R
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : I = ideal(a^2, a*b*(substitute(P,R)), b^2)
2 4 3 2 2 3 4
o5 = ideal (a , - 34a*b*c + 41a*b*c d + 30a*b*c d + 7a*b*c*d - 36a*b*d ,
------------------------------------------------------------------------
2
b )
o5 : Ideal of R</pre>
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<div>There is one minimal associated prime (a thick line in <i>P<sup>3</sup></i>) and <i>D</i> embedded primes (points on the line).</div>
<table class="examples"><tr><td><pre>i6 : ass I
o6 = {ideal (b, a), ideal (c - 13d, b, a), ideal (c - 39d, b, a), ideal (c -
------------------------------------------------------------------------
11d, b, a), ideal (c + 41d, b, a)}
o6 : List</pre>
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<tr><td><pre>i7 : primaryDecomposition I
2 2 2 2 2 2
o7 = {ideal (b , a*b, a ), ideal (c - 13d, b , a ), ideal (c + 41d, b , a ),
------------------------------------------------------------------------
2 2 2 2
ideal (c - 11d, b , a ), ideal (c - 39d, b , a )}
o7 : List</pre>
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<div>Only the minimal prime is a distinguished component, and it has multiplicity 2.</div>
<table class="examples"><tr><td><pre>i8 : distinguished(I)
o8 = {ideal (b, a)}
o8 : List</pre>
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<tr><td><pre>i9 : K = distinguishedAndMult(I)
o9 = {{2, ideal (b, a)}}
o9 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_distinguished__And__Mult.html" title="compute the distinguished subvarieties of a scheme along with their multiplicities">distinguishedAndMult</a> -- compute the distinguished subvarieties of a scheme along with their multiplicities</span></li>
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<div class="waystouse"><h2>Ways to use <tt>distinguished</tt> :</h2>
<ul><li>distinguished(Ideal)</li>
<li>distinguished(Ideal,RingElement)</li>
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