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<head><title>arrangement -- create a hyperplane arrangement</title>
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<div><h1>arrangement -- create a hyperplane arrangement</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>arrangement(L,R) or arrangement(M) or arrangement(M,R)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, a list of linear equations in the ring <tt>R</tt></span></li>
<li><span><tt>R</tt>, a polynomial ring or linear quotient of a polynomial ring</span></li>
<li><span><tt>M</tt>, a matrix whose rows represent linear forms defining hyperplanes</span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, the hyperplane arrangement determined by <tt>L</tt> and <tt>R</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>A hyperplane is a linear subspace of codimension one. An arrangement is a finite set of hyperplanes.<p/>
Probably the best-known hyperplane arrangement is the braid arrangement consisting of all the diagonal hyperplanes. In 4-space, it is constructed as follows: <table class="examples"><tr><td><pre>i1 : S = ZZ[w,x,y,z];</pre>
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<tr><td><pre>i2 : A3 = arrangement {w-x,w-y,w-z,x-y,x-z,y-z}
o2 = {w - x, w - y, w - z, x - y, x - z, y - z}
o2 : Hyperplane Arrangement </pre>
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<tr><td><pre>i3 : describe A3
o3 = {w - x, w - y, w - z, x - y, x - z, y - z}</pre>
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If we project along onto a subspace, then we obtain an essential arrangement:<table class="examples"><tr><td><pre>i4 : R = S/ideal(w+x+y+z)
o4 = R
o4 : QuotientRing</pre>
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<tr><td><pre>i5 : A3' = arrangement({w-x,w-y,w-z,x-y,x-z,y-z},R)
o5 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}
o5 : Hyperplane Arrangement </pre>
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<tr><td><pre>i6 : describe A3'
o6 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}</pre>
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The trivial arrangement has no equations.<table class="examples"><tr><td><pre>i7 : trivial = arrangement({},S)
o7 = {}
o7 : Hyperplane Arrangement </pre>
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<tr><td><pre>i8 : describe trivial
o8 = {}</pre>
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<tr><td><pre>i9 : ring trivial
o9 = S
o9 : PolynomialRing</pre>
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<div class="single"><h2>Caveat</h2>
<div>If the elements of <tt>L</tt> are not <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring elements</a> in <tt>R</tt>, then the induced identity map is used to map them from <tt>ring L#0</tt> into <tt>R</tt>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="hyperplane arrangements">HyperplaneArrangements</a> -- hyperplane arrangements</span></li>
<li><span><a href="_arrangement_lp__String_cm__Polynomial__Ring_rp.html" title="look up a built-in hyperplane arrangement">arrangement(String,PolynomialRing)</a> -- look up a built-in hyperplane arrangement</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>arrangement</tt> :</h2>
<ul><li>arrangement(Arrangement,Ring)</li>
<li>arrangement(List)</li>
<li>arrangement(List,Ring)</li>
<li>arrangement(Matrix)</li>
<li>arrangement(Matrix,Ring)</li>
<li><span><tt>arrangement(Flat)</tt> (missing documentation<!-- tag: (arrangement,Flat) -->)</span></li>
<li><span><tt>arrangement(String)</tt> (missing documentation<!-- tag: (arrangement,String) -->)</span></li>
<li><span><a href="_arrangement_lp__String_cm__Polynomial__Ring_rp.html" title="look up a built-in hyperplane arrangement">arrangement(String,PolynomialRing)</a> -- look up a built-in hyperplane arrangement</span></li>
<li><span><tt>arrangement(String,Ring)</tt> (missing documentation<!-- tag: (arrangement,String,Ring) -->)</span></li>
</ul>
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