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<!DOCTYPE html> <html lang="en"> <head> <title>toMap -- rational map defined by a linear system</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="package for some computations on rational maps between projective varieties" href="index.html">Cremona</a> :: <a title="rational map defined by a linear system" href="_to__Map.html">toMap</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | <a href="_to__External__String_lp__Rational__Map_rp.html">previous</a> | forward | <a href="_to__External__String_lp__Rational__Map_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>toMap -- rational map defined by a linear system</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">toMap("linear system")</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, or <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, etc.</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a href="___Dominant.html">Dominant</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span>, <span></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all ring maps" href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> <p>When the input represents a list of homogeneous elements $F_0,\ldots,F_m\in R=K[t_0,\ldots,t_n]/I$ of the same degree, then the method returns the ring map $\phi:K[x_0,\ldots,x_m] \to R$ that sends $x_i$ into $F_i$.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : QQ[t_0,t_1];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : linSys=gens (ideal(t_0,t_1))^5 o2 = | t_0^5 t_0^4t_1 t_0^3t_1^2 t_0^2t_1^3 t_0t_1^4 t_1^5 | 1 6 o2 : Matrix (QQ[t ..t ]) <-- (QQ[t ..t ]) 0 1 0 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : phi=toMap linSys 5 4 3 2 2 3 4 5 o3 = map (QQ[t ..t ], QQ[x ..x ], {t , t t , t t , t t , t t , t }) 0 1 0 5 0 0 1 0 1 0 1 0 1 1 o3 : RingMap QQ[t ..t ] <-- QQ[x ..x ] 0 1 0 5</code></pre> </td> </tr> </table> <p>If a positive integer $d$ is passed to the option <a href="___Dominant.html">Dominant</a>, then the method returns the induced map on $K[x_0,\ldots,x_m]/J_d$, where $J_d$ is the ideal generated by all homogeneous elements of degree $d$ of the kernel of $\phi$ (in this case <a title="homogeneous components of the kernel of a homogeneous ring map" href="_kernel_lp__Ring__Map_cm__Z__Z_rp.html">kernel(RingMap,ZZ)</a> is called).</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : phi'=toMap(linSys,Dominant=>2) QQ[x ..x ] 0 5 5 4 3 2 2 3 4 5 o4 = map (QQ[t ..t ], --------------------------------------------------------------------------------------------------------------------------, {t , t t , t t , t t , t t , t }) 0 1 2 2 2 2 0 0 1 0 1 0 1 0 1 1 (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x ) 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 QQ[x ..x ] 0 5 o4 : RingMap QQ[t ..t ] <-- -------------------------------------------------------------------------------------------------------------------------- 0 1 2 2 2 2 (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x ) 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2</code></pre> </td> </tr> </table> <p>If the input is a pair consisting of a homogeneous ideal $I$ and an integer $v$, then the output will be the map defined by the linear system of hypersurfaces of degree $v$ which contain the projective subscheme defined by $I$.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : I=kernel phi 2 2 o5 = ideal (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 ------------------------------------------------------------------------ 2 2 - x x , x x - x x , x - x x , x x - x x , x - x x ) 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o5 : Ideal of QQ[x ..x ] 0 5</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : toMap(I,2) 2 2 2 2 o6 = map (QQ[x ..x ], QQ[y ..y ], {x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x }) 0 5 0 9 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o6 : RingMap QQ[x ..x ] <-- QQ[y ..y ] 0 5 0 9</code></pre> </td> </tr> </table> <p>This is identical to <span class="tt">toMap(I,v,1)</span>, while the output of <span class="tt">toMap(I,v,e)</span> will be the map defined by the linear system of hypersurfaces of degree $v$ having points of multiplicity $e$ along the projective subscheme defined by $I$.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : toMap(I,2,1) 2 2 2 2 o7 = map (QQ[x ..x ], QQ[y ..y ], {x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x }) 0 5 0 9 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o7 : RingMap QQ[x ..x ] <-- QQ[y ..y ] 0 5 0 9</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : toMap(I,2,2) o8 = map (QQ[x ..x ], QQ[], {}) 0 5 o8 : RingMap QQ[x ..x ] <-- QQ[] 0 5</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : toMap(I,3,2) 3 2 2 2 2 2 2 2 2 3 2 2 o9 = map (QQ[x ..x ], QQ[y ..y ], {x - 2x x x + x x + x x - x x x , x x - x x - x x x + x x + x x x - x x x , x x - x x - x x x + x x x + x x - x x x , x - 2x x x + x x + x x - x x x }) 0 5 0 3 3 2 3 4 1 4 2 5 1 3 5 2 3 2 4 1 3 4 0 4 1 2 5 0 3 5 2 3 1 3 1 2 4 0 3 4 1 5 0 2 5 2 1 2 3 0 3 1 4 0 2 4 o9 : RingMap QQ[x ..x ] <-- QQ[y ..y ] 0 5 0 3</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="makes a rational map" href="_rational__Map.html">rationalMap(Matrix)</a> -- makes a rational map</span></li> <li><span><a title="makes a rational map from an ideal" href="_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html">rationalMap(Ideal,ZZ,ZZ)</a> -- makes a rational map from an ideal</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">toMap</span>:</h2> <ul> <li><kbd>toMap(Ideal)</kbd></li> <li><kbd>toMap(Ideal,List)</kbd></li> <li><kbd>toMap(Ideal,ZZ)</kbd></li> <li><kbd>toMap(Ideal,ZZ,ZZ)</kbd></li> <li><kbd>toMap(List)</kbd></li> <li><kbd>toMap(Matrix)</kbd></li> <li><kbd>toMap(RingMap)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="rational map defined by a linear system" href="_to__Map.html">toMap</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Cremona/documentation.m2:846:0</span>.</p> </div> </div> </div> </body> </html>