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<!DOCTYPE html> <html lang="en"> <head> <title>Example 3 -- jets of determinantal varieties</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="compute jets of various algebraic, geometric and combinatorial objects" href="index.html">Jets</a> » <a title="jets of determinantal varieties" href="___Example_sp3.html">Example 3</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> <a href="___Example_sp4.html">next</a> | <a href="___Example_sp2.html">previous</a> | <a href="___Example_sp4.html">forward</a> | <a href="___Example_sp2.html">backward</a> | <a href="index.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Example 3 -- jets of determinantal varieties</h1> <div> <div> <p>Consider the determinantal varieties $X_r$ of $3\times 3$ matrices of rank at most $r$, which are defined by the vanishing of minors of size $r+1$. We illustrate computationally some of the known results about jets.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x_(1,1)..x_(3,3)] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : G = genericMatrix(R,3,3) o2 = | x_(1,1) x_(2,1) x_(3,1) | | x_(1,2) x_(2,2) x_(3,2) | | x_(1,3) x_(2,3) x_(3,3) | 3 3 o2 : Matrix R <-- R</code></pre> </td> </tr> </table> <div> <p>Since $X_0$ is a single point, its first jet scheme consists of a single (smooth) point.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : I1 = minors(1,G) o3 = ideal (x , x , x , x , x , x , x , x , x ) 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 o3 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : JI1 = jets(1,I1) o4 = ideal (x1 , x0 , x1 , x0 , x1 , x0 , x1 , x0 , x1 , 1,1 1,1 1,2 1,2 1,3 1,3 2,1 2,1 2,2 ------------------------------------------------------------------------ x0 , x1 , x0 , x1 , x0 , x1 , x0 , x1 , x0 ) 2,2 2,3 2,3 3,1 3,1 3,2 3,2 3,3 3,3 o4 : Ideal of QQ[x0 ..x0 ][x1 ..x1 ] 1,1 3,3 1,1 3,3</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : dim JI1, isPrime JI1 o5 = (0, true) o5 : Sequence</code></pre> </td> </tr> </table> <div> <p>The jets of $X_2$ (the determinantal hypersurface) are known to be irreducible (see Theorem 3.1 in <a href="https://doi.org/10.1016/j.jpaa.2004.06.001">T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings</a> [KS05], or Corollary 4.13 in <a href="https://doi.org/10.1090/S0002-9947-2012-05564-4">R. Docampo, Arcs on determinantal varieties</a> [Doc13]). Since $X_2$ is a complete intersection and has rational singularities (see Corollary 6.1.5(b) in <a href="https://doi.org/10.1017/CBO9780511546556">J. Weyman, Cohomology of vector bundles and syzygies</a>), this also follows from a more general result of M. Mustaţă (Theorem 3.3 in <a href="https://doi.org/10.1007/s002220100152">Jet schemes of locally complete intersection canonical singularities</a>).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : I3 = minors(3,G) o6 = ideal(- x x x + x x x + x x x - x x x - 1,3 2,2 3,1 1,2 2,3 3,1 1,3 2,1 3,2 1,1 2,3 3,2 ------------------------------------------------------------------------ x x x + x x x ) 1,2 2,1 3,3 1,1 2,2 3,3 o6 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : JI3 = jets(1,I3) o7 = ideal ((- x0 x0 + x0 x0 )x1 + (x0 x0 - x0 x0 )x1 2,3 3,2 2,2 3,3 1,1 2,3 3,1 2,1 3,3 1,2 ------------------------------------------------------------------------ + (- x0 x0 + x0 x0 )x1 + (x0 x0 - x0 x0 )x1 + (- 2,2 3,1 2,1 3,2 1,3 1,3 3,2 1,2 3,3 2,1 ------------------------------------------------------------------------ x0 x0 + x0 x0 )x1 + (x0 x0 - x0 x0 )x1 + (- 1,3 3,1 1,1 3,3 2,2 1,2 3,1 1,1 3,2 2,3 ------------------------------------------------------------------------ x0 x0 + x0 x0 )x1 + (x0 x0 - x0 x0 )x1 + (- 1,3 2,2 1,2 2,3 3,1 1,3 2,1 1,1 2,3 3,2 ------------------------------------------------------------------------ x0 x0 + x0 x0 )x1 , - x0 x0 x0 + x0 x0 x0 + 1,2 2,1 1,1 2,2 3,3 1,3 2,2 3,1 1,2 2,3 3,1 ------------------------------------------------------------------------ x0 x0 x0 - x0 x0 x0 - x0 x0 x0 + x0 x0 x0 ) 1,3 2,1 3,2 1,1 2,3 3,2 1,2 2,1 3,3 1,1 2,2 3,3 o7 : Ideal of QQ[x0 ..x0 ][x1 ..x1 ] 1,1 3,3 1,1 3,3</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : isPrime JI3 o8 = true</code></pre> </td> </tr> </table> <div> <p>As for the case of $2\times 2$ minors, Theorem 5.1 in [KS05], Theorem 5.1 in <a href="https://arxiv.org/abs/math/0608632">C. Yuen, Jet schemes of determinantal varieties</a>, and Corollary 4.13 in [Doc13] all count the number of components; the first two references describe the components further. As expected, the first jet scheme of $X_1$ has two components, one of them an affine space.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : I2 = minors(2,G) o9 = ideal (- x x + x x , - x x + x x , - x x + 1,2 2,1 1,1 2,2 1,3 2,1 1,1 2,3 1,3 2,2 ------------------------------------------------------------------------ x x , - x x + x x , - x x + x x , - x x + 1,2 2,3 1,2 3,1 1,1 3,2 1,3 3,1 1,1 3,3 1,3 3,2 ------------------------------------------------------------------------ x x , - x x + x x , - x x + x x , - x x + 1,2 3,3 2,2 3,1 2,1 3,2 2,3 3,1 2,1 3,3 2,3 3,2 ------------------------------------------------------------------------ x x ) 2,2 3,3 o9 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : JI2 = jets(1,I2) o10 = ideal (x0 x1 - x0 x1 - x0 x1 + x0 x1 , - x0 x0 2,2 1,1 2,1 1,2 1,2 2,1 1,1 2,2 1,2 2,1 ----------------------------------------------------------------------- + x0 x0 , x0 x1 - x0 x1 - x0 x1 + x0 x1 , - 1,1 2,2 2,3 1,1 2,1 1,3 1,3 2,1 1,1 2,3 ----------------------------------------------------------------------- x0 x0 + x0 x0 , x0 x1 - x0 x1 - x0 x1 + 1,3 2,1 1,1 2,3 2,3 1,2 2,2 1,3 1,3 2,2 ----------------------------------------------------------------------- x0 x1 , - x0 x0 + x0 x0 , x0 x1 - x0 x1 - 1,2 2,3 1,3 2,2 1,2 2,3 3,2 1,1 3,1 1,2 ----------------------------------------------------------------------- x0 x1 + x0 x1 , - x0 x0 + x0 x0 , x0 x1 - 1,2 3,1 1,1 3,2 1,2 3,1 1,1 3,2 3,3 1,1 ----------------------------------------------------------------------- x0 x1 - x0 x1 + x0 x1 , - x0 x0 + x0 x0 , 3,1 1,3 1,3 3,1 1,1 3,3 1,3 3,1 1,1 3,3 ----------------------------------------------------------------------- x0 x1 - x0 x1 - x0 x1 + x0 x1 , - x0 x0 + 3,3 1,2 3,2 1,3 1,3 3,2 1,2 3,3 1,3 3,2 ----------------------------------------------------------------------- x0 x0 , x0 x1 - x0 x1 - x0 x1 + x0 x1 , - 1,2 3,3 3,2 2,1 3,1 2,2 2,2 3,1 2,1 3,2 ----------------------------------------------------------------------- x0 x0 + x0 x0 , x0 x1 - x0 x1 - x0 x1 + 2,2 3,1 2,1 3,2 3,3 2,1 3,1 2,3 2,3 3,1 ----------------------------------------------------------------------- x0 x1 , - x0 x0 + x0 x0 , x0 x1 - x0 x1 - 2,1 3,3 2,3 3,1 2,1 3,3 3,3 2,2 3,2 2,3 ----------------------------------------------------------------------- x0 x1 + x0 x1 , - x0 x0 + x0 x0 ) 2,3 3,2 2,2 3,3 2,3 3,2 2,2 3,3 o10 : Ideal of QQ[x0 ..x0 ][x1 ..x1 ] 1,1 3,3 1,1 3,3</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : P = primaryDecomposition JI2; #P o12 = 2</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : P_1 o13 = ideal (x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 ) 3,3 3,2 3,1 2,3 2,2 2,1 1,3 1,2 1,1 o13 : Ideal of QQ[x0 ..x0 ][x1 ..x1 ] 1,1 3,3 1,1 3,3</code></pre> </td> </tr> </table> <div> <p>The other component is the so-called principal component of the jet scheme, i.e., the Zariski closure of the first jets of the smooth locus of $X_1$. To check this, we first establish that the first jet scheme is reduced (i.e. its ideal is radical), then use the <a title="compute principal component of jets" href="_principal__Component.html">principalComponent</a> method with the option <a title="option for principal components" href="___Saturate.html">principalComponent(...,Saturate=>...)</a> set to <span class="tt">false</span> to speed up computations.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i14 : radical JI2 == JI2 o14 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : P_0 == principalComponent(1,I2,Saturate=>false) o15 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : P_0 o16 = ideal (x0 x0 - x0 x0 , x0 x0 - x0 x0 , x0 x0 - 2,3 3,2 2,2 3,3 1,3 3,2 1,2 3,3 2,3 3,1 ----------------------------------------------------------------------- x0 x0 , x0 x0 - x0 x0 , x0 x0 - x0 x0 , 2,1 3,3 2,2 3,1 2,1 3,2 1,3 3,1 1,1 3,3 ----------------------------------------------------------------------- x0 x0 - x0 x0 , x0 x0 - x0 x0 , x0 x0 - 1,2 3,1 1,1 3,2 1,3 2,2 1,2 2,3 1,3 2,1 ----------------------------------------------------------------------- x0 x0 , x0 x0 - x0 x0 , x0 x1 - x0 x1 - 1,1 2,3 1,2 2,1 1,1 2,2 3,3 2,2 3,2 2,3 ----------------------------------------------------------------------- x0 x1 + x0 x1 , x0 x1 - x0 x1 - x0 x1 + 2,3 3,2 2,2 3,3 3,3 2,1 3,1 2,3 2,3 3,1 ----------------------------------------------------------------------- x0 x1 , x0 x1 - x0 x1 - x0 x1 + x0 x1 , 2,1 3,3 3,2 2,1 3,1 2,2 2,2 3,1 2,1 3,2 ----------------------------------------------------------------------- x0 x1 - x0 x1 - x0 x1 + x0 x1 , x0 x1 - 3,3 1,2 3,2 1,3 1,3 3,2 1,2 3,3 2,3 1,2 ----------------------------------------------------------------------- x0 x1 - x0 x1 + x0 x1 , x0 x1 - x0 x1 - 2,2 1,3 1,3 2,2 1,2 2,3 3,3 1,1 3,1 1,3 ----------------------------------------------------------------------- x0 x1 + x0 x1 , x0 x1 - x0 x1 - x0 x1 + 1,3 3,1 1,1 3,3 3,2 1,1 3,1 1,2 1,2 3,1 ----------------------------------------------------------------------- x0 x1 , x0 x1 - x0 x1 - x0 x1 + x0 x1 , 1,1 3,2 2,3 1,1 2,1 1,3 1,3 2,1 1,1 2,3 ----------------------------------------------------------------------- x0 x1 - x0 x1 - x0 x1 + x0 x1 , x1 x1 x1 - 2,2 1,1 2,1 1,2 1,2 2,1 1,1 2,2 1,3 2,2 3,1 ----------------------------------------------------------------------- x1 x1 x1 - x1 x1 x1 + x1 x1 x1 + x1 x1 x1 - 1,2 2,3 3,1 1,3 2,1 3,2 1,1 2,3 3,2 1,2 2,1 3,3 ----------------------------------------------------------------------- x1 x1 x1 ) 1,1 2,2 3,3 o16 : Ideal of QQ[x0 ..x0 ][x1 ..x1 ] 1,1 3,3 1,1 3,3</code></pre> </td> </tr> </table> <div> <p>Finally, as observed in Theorem 18 of <a href="http://dx.doi.org/10.2140/pjm.2014.272.147">S.R. Ghorpade, B. Jonov and B.A. Sethuraman, Hilbert series of certain jet schemes of determinantal varieties</a> the Hilbert series of the principal component of the first jet scheme of $X_1$ is the square of the Hilbert series of $X_1$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i17 : apply({P_0,I2}, X -> hilbertSeries(X,Reduce=>true)) 2 3 4 2 1 + 8T + 18T + 8T + T 1 + 4T + T o17 = {------------------------, -----------} 10 5 (1 - T) (1 - T) o17 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : numerator (first oo) == (numerator last oo)^2 o18 = true</code></pre> </td> </tr> </table> </div> <div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Jets.m2:1666:0</span>.</p> </div> </div> </div> </body> </html>