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<head><title>specialFiber -- special fiber of a blowup</title>
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<div><h1>specialFiber -- special fiber of a blowup</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>specialFiber M</tt><br/><tt>specialFiber(M,f)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, or <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>, an optional element, which is a non-zerodivisor modulo <tt>M</tt> and the ring of <tt>M</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</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>M</i> be an <i>R = k[x<sub>1</sub>,...,x<sub>n</sub>]/J</i>-module (for example an ideal), and let <i>mm=ideal vars R = (x<sub>1</sub>,...,x<sub>n</sub>)</i>, and suppose that <i>M</i> is a homomorphic image of the free module <i>F</i> with <i>m+1</i> generators. Let <i>T</i> be the Rees algebra of <i>M</i>. The call specialFiber(M) returns the ideal <i>J⊂k[w<sub>0</sub>,…,w<sub>m</sub>]</i> such that <i>k[w<sub>0</sub>,…,w<sub>m</sub>]/J ≅T/mm*T</i>; that is, specialFiber(M) = reesIdeal(M)+mm*Sym(F). This routine differs from <a href="_special__Fiber__Ideal.html" title="special fiber of a blowup">specialFiberIdeal</a> in that the ambient ring of the output ideal is <i>k[w<sub>0</sub>,…,w<sub>m</sub>]</i> rather than <i>R[w<sub>0</sub>,…,w<sub>m</sub>]</i>. The coefficient ring <i>k</i> used is always the <a href="../../Macaulay2Doc/html/_ultimate.html" title="ultimate value for an iteration">ultimate</a> <a href="../../Macaulay2Doc/html/_coefficient__Ring.html" title="get the coefficient ring">coefficientRing</a> of <i>R</i>.</p>
<div>The name derives from the fact that <i>Proj(T/mm*T)</i> is the special fiber of the blowup of <i>Spec R</i> along the subscheme defined by <i>I</i>.</div>
<table class="examples"><tr><td><pre>i1 : R=QQ[a,b,c,d,e,f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M=matrix{{a,c,e},{b,d,f}}
o2 = | a c e |
| b d f |
2 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : analyticSpread image M
o3 = 3</pre>
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<tr><td><pre>i4 : specialFiber image M
o4 = QQ[w , w , w ]
0 1 2
o4 : PolynomialRing</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_rees__Ideal.html" title="compute the defining ideal of the Rees Algebra">reesIdeal</a> -- compute the defining ideal of the Rees Algebra</span></li>
<li><span><a href="_special__Fiber__Ideal.html" title="special fiber of a blowup">specialFiberIdeal</a> -- special fiber of a blowup</span></li>
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<div class="waystouse"><h2>Ways to use <tt>specialFiber</tt> :</h2>
<ul><li>specialFiber(Ideal)</li>
<li>specialFiber(Ideal,RingElement)</li>
<li>specialFiber(Module)</li>
<li>specialFiber(Module,RingElement)</li>
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