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<!DOCTYPE html> <html lang="en"> <head> <title>Divisor -- divisors</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="divisors" href="index.html">Divisor</a> :: <a title="divisors" href="index.html">Divisor</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="___Ambient__Ring.html">next</a> | previous | <a href="___Ambient__Ring.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Divisor -- divisors</h1> <div> <h2>Description</h2> <em>Divisor</em> is a package for working with (Q/R)-Weil divisors on <em>normal</em> affine and projective varieties (equivalently, on commutative, normal and graded rings).<br><br>This package introduces a type <a title="the Types of divisors" href="___Basic__Divisor.html">WeilDivisor</a> which lets the user work with Weil divisors similar to the way one might in algebraic geometry. We highlight a few important functions below.<br><br><b>Useful functions:</b><br> <ul> <li><a title="whether a Weil divisor is Cartier" href="_is__Cartier.html">isCartier</a> or <a title="whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1." href="_is__Q__Cartier.html">isQCartier</a> can let you determine if a divisor is Cartier or if a power is Cartier.</li> <li><a href="../../Macaulay2Doc/html/_is__Very__Ample.html">isVeryAmple</a> lets you check if a divisor is very ample.</li> <li><a title="compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated" href="_base__Locus.html">baseLocus</a> lets you compute the base locus of the complete linear system corresponding to a divisor on a projective variety.</li> <li><a title="compute the map to projective space associated with the global sections of a Cartier divisor" href="_map__To__Projective__Space.html">mapToProjectiveSpace</a> returns the map to projective space determined by the complete linear system determined by the divisor.</li> <li><a title="compute a canonical divisor of a ring" href="_canonical__Divisor.html">canonicalDivisor</a> lets you compute the canonical divisor on some affine or projective variety.</li> <li><a title="compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base" href="_ramification__Divisor.html">ramificationDivisor</a> lets you compute the relative canonical divisor of a finite map varieties.</li> </ul> <br>This package also includes some functions for interacting with ideals and modules which might be independently useful.<br> <ul> <li><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> embeds a rank one module as an ideal.</li> <li><a title="calculate the double dual of an ideal or module Hom(Hom(M, R), R)" href="_reflexify.html">reflexify</a> computes the reflexification, Hom(Hom(M, R), R) of a module M or ideal.</li> <li><a title="computes a reflexive power of an ideal in a normal domain" href="_reflexive__Power.html">reflexivePower</a> computes the reflexification of a power of an ideal quickly.</li> <li><a title="create the torsion submodule of a module" href="_torsion__Submodule.html">torsionSubmodule</a> find the torsion submodule of a module.</li> </ul> <br>We emphasize once more that the functions in this package might produce unexpected results on non-normal rings.<br><br><b>Acknowledgements:</b><br><br>The authors would like to thank Tommaso de Fernex, David Eisenbud, Daniel Grayson, Anurag Singh, Greg Smith, Mike Stillman and the referee for useful conversations and comments on the development of this package.<br> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.math.utah.edu/~schwede">Karl Schwede</a><span> <<a href="mailto:kschwede%40gmail.com">kschwede@gmail.com</a>></span></li> <li>Zhaoning Yang<span> <<a href="mailto:zyy5054%40gmail.com">zyy5054@gmail.com</a>></span></li> </ul> </div> <div> <h2>Certification <img src="../../../../Macaulay2/Style/GoldStar.png" alt="a gold star"> </h2> <p>Version <b>0.3</b> of this package was accepted for publication in <a href="https://msp.org/jsag/2018/8-1/">volume 8</a> of <a href="https://msp.org/jsag/">The Journal of Software for Algebra and Geometry</a> on 31 August 2018, in the article <a href="https://msp.org/jsag/2018/8-1/p09.xhtml">Divisor Package for Macaulay2</a> (DOI: <a href="https://doi.org/10.2140/jsag.2018.8.87">10.2140/jsag.2018.8.87</a>). That version can be obtained from <a href="https://msp.org/jsag/2018/8-1/jsag-v8-n1-x09-Divisor.m2">the journal</a>.</p> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>0.3</b> of Divisor, released <b>May 30th, 2018</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{DivisorSource, title = {{Divisor: Weil divisors. Version~0.3}}, author = {Karl Schwede and Zhaoning Yang}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } @article{DivisorArticle, title = {{Divisor Package for \emph{Macaulay2}}}, author = {Karl Schwede and Zhaoning Yang}, journal = {The Journal of Software for Algebra and Geometry}, volume = {8}, year = {2018}, } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="the Types of divisors" href="___Basic__Divisor.html">BasicDivisor</a> -- the Types of divisors</span></li> <li><span><kbd>QWeilDivisor</kbd> -- see <span><a title="the Types of divisors" href="___Basic__Divisor.html">BasicDivisor</a> -- the Types of divisors</span></span></li> <li><span><kbd>RWeilDivisor</kbd> -- see <span><a title="the Types of divisors" href="___Basic__Divisor.html">BasicDivisor</a> -- the Types of divisors</span></span></li> <li><span><kbd>WeilDivisor</kbd> -- see <span><a title="the Types of divisors" href="___Basic__Divisor.html">BasicDivisor</a> -- the Types of divisors</span></span></li> </ul> </li> <li>Functions and commands <ul> <li><span><a title="apply a function to the coefficients of a divisor" href="_apply__To__Coefficients.html">applyToCoefficients</a> -- apply a function to the coefficients of a divisor</span></li> <li><span><a title="compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated" href="_base__Locus.html">baseLocus</a> -- compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated</span></li> <li><span><a title="compute a canonical divisor of a ring" href="_canonical__Divisor.html">canonicalDivisor</a> -- compute a canonical divisor of a ring</span></li> <li><span><a title="removes primes with coefficient zero from a divisor" href="_clean__Support.html">cleanSupport</a> -- removes primes with coefficient zero from a divisor</span></li> <li><span><a title="creates a new divisor with most entries from the cache removed" href="_clear__Cache.html">clearCache</a> -- creates a new divisor with most entries from the cache removed</span></li> <li><span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></li> <li><span><a title="finds an ideal or module isomorphic to Hom(M, R)" href="_dualize.html">dualize</a> -- finds an ideal or module isomorphic to Hom(M, R)</span></li> <li><span><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> -- embed a module as an ideal of a ring</span></li> <li><span><a title="find an element of a specified degree" href="_find__Element__Of__Degree.html">findElementOfDegree</a> -- find an element of a specified degree</span></li> <li><span><a title="get the list of Groebner bases corresponding to the height-one primes in the support of a divisor" href="_gbs.html">gbs</a> -- get the list of Groebner bases corresponding to the height-one primes in the support of a divisor</span></li> <li><span><a title="find a solution of the linear Diophantine equation Ax = b" href="_get__Linear__Diophantine__Solution.html">getLinearDiophantineSolution</a> -- find a solution of the linear Diophantine equation Ax = b</span></li> <li><span><a title="get the number of height-one primes in the support of the divisor" href="_get__Prime__Count.html">getPrimeCount</a> -- get the number of height-one primes in the support of the divisor</span></li> <li><span><a title="get the list of prime divisors of a given divisor" href="_get__Prime__Divisors.html">getPrimeDivisors</a> -- get the list of prime divisors of a given divisor</span></li> <li><span><a title="compute the ideal generated by the generators of the ideal raised to a power" href="_ideal__Power.html">idealPower</a> -- compute the ideal generated by the generators of the ideal raised to a power</span></li> <li><span><a title="whether a Weil divisor is Cartier" href="_is__Cartier.html">isCartier</a> -- whether a Weil divisor is Cartier</span></li> <li><span><a title="whether a ring is a domain" href="_is__Domain.html">isDomain</a> -- whether a ring is a domain</span></li> <li><span><a title="whether a divisor is effective" href="_is__Effective.html">isEffective</a> -- whether a divisor is effective</span></li> <li><span><a title="whether two Weil divisors are linearly equivalent" href="_is__Linear__Equivalent.html">isLinearEquivalent</a> -- whether two Weil divisors are linearly equivalent</span></li> <li><span><a title="whether a Weil divisor is globally principal" href="_is__Principal.html">isPrincipal</a> -- whether a Weil divisor is globally principal</span></li> <li><span><a title="whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1." href="_is__Q__Cartier.html">isQCartier</a> -- whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.</span></li> <li><span><a title="whether two Q-divisors are linearly equivalent" href="_is__Q__Linear__Equivalent.html">isQLinearEquivalent</a> -- whether two Q-divisors are linearly equivalent</span></li> <li><span><a title="whether a divisor is reduced" href="_is__Reduced.html">isReduced</a> -- whether a divisor is reduced</span></li> <li><span><a title="whether the divisor is simple normal crossings" href="_is__S__N__C.html">isSNC</a> -- whether the divisor is simple normal crossings</span></li> <li><span><a title="whether a rational/real divisor is in actuality a Weil divisor" href="_is__Weil__Divisor.html">isWeilDivisor</a> -- whether a rational/real divisor is in actuality a Weil divisor</span></li> <li><span><a title="whether the divisor is the zero divisor" href="_is__Zero__Divisor.html">isZeroDivisor</a> -- whether the divisor is the zero divisor</span></li> <li><span><a title="compute the map to projective space associated with the global sections of a Cartier divisor" href="_map__To__Projective__Space.html">mapToProjectiveSpace</a> -- compute the map to projective space associated with the global sections of a Cartier divisor</span></li> <li><span><a title="the non-Cartier locus of a Weil divisor" href="_non__Cartier__Locus.html">nonCartierLocus</a> -- the non-Cartier locus of a Weil divisor</span></li> <li><span><kbd>negativePart</kbd> -- see <span><a title="get the effective part or anti-effective part of a divisor" href="_positive__Part.html">positivePart</a> -- get the effective part or anti-effective part of a divisor</span></span></li> <li><span><a title="get the effective part or anti-effective part of a divisor" href="_positive__Part.html">positivePart</a> -- get the effective part or anti-effective part of a divisor</span></li> <li><span><a title="get the list of height-one primes in the support of a divisor" href="_primes.html">primes</a> -- get the list of height-one primes in the support of a divisor</span></li> <li><span><a title="compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base" href="_ramification__Divisor.html">ramificationDivisor</a> -- compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base</span></li> <li><span><a title="calculate the double dual of an ideal or module Hom(Hom(M, R), R)" href="_reflexify.html">reflexify</a> -- calculate the double dual of an ideal or module Hom(Hom(M, R), R)</span></li> <li><span><a title="computes a reflexive power of an ideal in a normal domain" href="_reflexive__Power.html">reflexivePower</a> -- computes a reflexive power of an ideal in a normal domain</span></li> <li><span><a title="create a Q-Weil divisor from a Weil divisor" href="_to__Q__Weil__Divisor.html">toQWeilDivisor</a> -- create a Q-Weil divisor from a Weil divisor</span></li> <li><span><a title="create the torsion submodule of a module" href="_torsion__Submodule.html">torsionSubmodule</a> -- create the torsion submodule of a module</span></li> <li><span><a title="create a R-divisor from a Q or Weil divisor" href="_to__R__Weil__Divisor.html">toRWeilDivisor</a> -- create a R-divisor from a Q or Weil divisor</span></li> <li><span><a title="create a Weil divisor from a Q or R-divisor" href="_to__Weil__Divisor.html">toWeilDivisor</a> -- create a Weil divisor from a Q or R-divisor</span></li> <li><span><a title="constructs the zero Weil divisor for the ring" href="_zero__Divisor.html">zeroDivisor</a> -- constructs the zero Weil divisor for the ring</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>applyToCoefficients(BasicDivisor,Function)</kbd> -- see <span><a title="apply a function to the coefficients of a divisor" href="_apply__To__Coefficients.html">applyToCoefficients</a> -- apply a function to the coefficients of a divisor</span></span></li> <li><span><kbd>baseLocus(Module)</kbd> -- see <span><a title="compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated" href="_base__Locus.html">baseLocus</a> -- compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated</span></span></li> <li><span><kbd>baseLocus(WeilDivisor)</kbd> -- see <span><a title="compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated" href="_base__Locus.html">baseLocus</a> -- compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated</span></span></li> <li><span><kbd>- BasicDivisor</kbd> -- see <span><a title="add or subtract two divisors, or negate a divisor" href="___Basic__Divisor_sp_pl_sp__Basic__Divisor.html">BasicDivisor + BasicDivisor</a> -- add or subtract two divisors, or negate a divisor</span></span></li> <li><span><a title="add or subtract two divisors, or negate a divisor" href="___Basic__Divisor_sp_pl_sp__Basic__Divisor.html">BasicDivisor + BasicDivisor</a> -- add or subtract two divisors, or negate a divisor</span></li> <li><span><kbd>BasicDivisor - BasicDivisor</kbd> -- see <span><a title="add or subtract two divisors, or negate a divisor" href="___Basic__Divisor_sp_pl_sp__Basic__Divisor.html">BasicDivisor + BasicDivisor</a> -- add or subtract two divisors, or negate a divisor</span></span></li> <li><span><kbd>canonicalDivisor(Ring)</kbd> -- see <span><a title="compute a canonical divisor of a ring" href="_canonical__Divisor.html">canonicalDivisor</a> -- compute a canonical divisor of a ring</span></span></li> <li><span><a title="produce a WeilDivisor whose coefficients are ceilings or floors of the divisor" href="_ceiling_lp__R__Weil__Divisor_rp.html">ceiling(RWeilDivisor)</a> -- produce a WeilDivisor whose coefficients are ceilings or floors of the divisor</span></li> <li><span><kbd>floor(RWeilDivisor)</kbd> -- see <span><a title="produce a WeilDivisor whose coefficients are ceilings or floors of the divisor" href="_ceiling_lp__R__Weil__Divisor_rp.html">ceiling(RWeilDivisor)</a> -- produce a WeilDivisor whose coefficients are ceilings or floors of the divisor</span></span></li> <li><span><kbd>cleanSupport(BasicDivisor)</kbd> -- see <span><a title="removes primes with coefficient zero from a divisor" href="_clean__Support.html">cleanSupport</a> -- removes primes with coefficient zero from a divisor</span></span></li> <li><span><kbd>clearCache(BasicDivisor)</kbd> -- see <span><a title="creates a new divisor with most entries from the cache removed" href="_clear__Cache.html">clearCache</a> -- creates a new divisor with most entries from the cache removed</span></span></li> <li><span><a title="get the coefficient of an ideal for a fixed divisor" href="_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.html">coefficient(BasicList,BasicDivisor)</a> -- get the coefficient of an ideal for a fixed divisor</span></li> <li><span><a title="get the coefficient of an ideal for a fixed divisor" href="_coefficient_lp__Ideal_cm__Basic__Divisor_rp.html">coefficient(Ideal,BasicDivisor)</a> -- get the coefficient of an ideal for a fixed divisor</span></li> <li><span><a title="get the list of coefficients of a divisor" href="_coefficients_lp__Basic__Divisor_rp.html">coefficients(BasicDivisor)</a> -- get the list of coefficients of a divisor</span></li> <li><span><kbd>divisor(BasicList)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>divisor(BasicList,BasicList)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>divisor(Ideal)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>divisor(Matrix)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>divisor(Module)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>divisor(RingElement)</kbd> -- see <span><a title="constructor for (Weil/Q/R)-divisors" href="_divisor.html">divisor</a> -- constructor for (Weil/Q/R)-divisors</span></span></li> <li><span><kbd>dualize(Ideal)</kbd> -- see <span><a title="finds an ideal or module isomorphic to Hom(M, R)" href="_dualize.html">dualize</a> -- finds an ideal or module isomorphic to Hom(M, R)</span></span></li> <li><span><kbd>dualize(Module)</kbd> -- see <span><a title="finds an ideal or module isomorphic to Hom(M, R)" href="_dualize.html">dualize</a> -- finds an ideal or module isomorphic to Hom(M, R)</span></span></li> <li><span><kbd>embedAsIdeal(Matrix)</kbd> -- see <span><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> -- embed a module as an ideal of a ring</span></span></li> <li><span><kbd>embedAsIdeal(Module)</kbd> -- see <span><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> -- embed a module as an ideal of a ring</span></span></li> <li><span><kbd>embedAsIdeal(Ring,Matrix)</kbd> -- see <span><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> -- embed a module as an ideal of a ring</span></span></li> <li><span><kbd>embedAsIdeal(Ring,Module)</kbd> -- see <span><a title="embed a module as an ideal of a ring" href="_embed__As__Ideal.html">embedAsIdeal</a> -- embed a module as an ideal of a ring</span></span></li> <li><span><kbd>findElementOfDegree(BasicList,Ring)</kbd> -- see <span><a title="find an element of a specified degree" href="_find__Element__Of__Degree.html">findElementOfDegree</a> -- find an element of a specified degree</span></span></li> <li><span><kbd>findElementOfDegree(ZZ,Ring)</kbd> -- see <span><a title="find an element of a specified degree" href="_find__Element__Of__Degree.html">findElementOfDegree</a> -- find an element of a specified degree</span></span></li> <li><span><kbd>gbs(BasicDivisor)</kbd> -- see <span><a title="get the list of Groebner bases corresponding to the height-one primes in the support of a divisor" href="_gbs.html">gbs</a> -- get the list of Groebner bases corresponding to the height-one primes in the support of a divisor</span></span></li> <li><span><kbd>getLinearDiophantineSolution(BasicList,BasicList)</kbd> -- see <span><a title="find a solution of the linear Diophantine equation Ax = b" href="_get__Linear__Diophantine__Solution.html">getLinearDiophantineSolution</a> -- find a solution of the linear Diophantine equation Ax = b</span></span></li> <li><span><kbd>getLinearDiophantineSolution(BasicList,Matrix)</kbd> -- see <span><a title="find a solution of the linear Diophantine equation Ax = b" href="_get__Linear__Diophantine__Solution.html">getLinearDiophantineSolution</a> -- find a solution of the linear Diophantine equation Ax = b</span></span></li> <li><span><kbd>getPrimeCount(BasicDivisor)</kbd> -- see <span><a title="get the number of height-one primes in the support of the divisor" href="_get__Prime__Count.html">getPrimeCount</a> -- get the number of height-one primes in the support of the divisor</span></span></li> <li><span><kbd>getPrimeDivisors(BasicDivisor)</kbd> -- see <span><a title="get the list of prime divisors of a given divisor" href="_get__Prime__Divisors.html">getPrimeDivisors</a> -- get the list of prime divisors of a given divisor</span></span></li> <li><span><kbd>ideal(QWeilDivisor)</kbd> -- see <span><a title="calculate the corresponding module of a divisor and represent it as an ideal" href="_ideal_lp__R__Weil__Divisor_rp.html">ideal(RWeilDivisor)</a> -- calculate the corresponding module of a divisor and represent it as an ideal</span></span></li> <li><span><a title="calculate the corresponding module of a divisor and represent it as an ideal" href="_ideal_lp__R__Weil__Divisor_rp.html">ideal(RWeilDivisor)</a> -- calculate the corresponding module of a divisor and represent it as an ideal</span></li> <li><span><kbd>ideal(WeilDivisor)</kbd> -- see <span><a title="calculate the corresponding module of a divisor and represent it as an ideal" href="_ideal_lp__R__Weil__Divisor_rp.html">ideal(RWeilDivisor)</a> -- calculate the corresponding module of a divisor and represent it as an ideal</span></span></li> <li><span><kbd>idealPower(ZZ,Ideal)</kbd> -- see <span><a title="compute the ideal generated by the generators of the ideal raised to a power" href="_ideal__Power.html">idealPower</a> -- compute the ideal generated by the generators of the ideal raised to a power</span></span></li> <li><span><kbd>isCartier(WeilDivisor)</kbd> -- see <span><a title="whether a Weil divisor is Cartier" href="_is__Cartier.html">isCartier</a> -- whether a Weil divisor is Cartier</span></span></li> <li><span><kbd>isDomain(Ring)</kbd> -- see <span><a title="whether a ring is a domain" href="_is__Domain.html">isDomain</a> -- whether a ring is a domain</span></span></li> <li><span><kbd>isEffective(BasicDivisor)</kbd> -- see <span><a title="whether a divisor is effective" href="_is__Effective.html">isEffective</a> -- whether a divisor is effective</span></span></li> <li><span><a title="whether the divisor is graded (homogeneous)" href="_is__Homogeneous_lp__Basic__Divisor_rp.html">isHomogeneous(BasicDivisor)</a> -- whether the divisor is graded (homogeneous)</span></li> <li><span><kbd>isLinearEquivalent(WeilDivisor,WeilDivisor)</kbd> -- see <span><a title="whether two Weil divisors are linearly equivalent" href="_is__Linear__Equivalent.html">isLinearEquivalent</a> -- whether two Weil divisors are linearly equivalent</span></span></li> <li><span><a title="whether a divisor is prime" href="_is__Prime_lp__Basic__Divisor_rp.html">isPrime(BasicDivisor)</a> -- whether a divisor is prime</span></li> <li><span><kbd>isPrincipal(WeilDivisor)</kbd> -- see <span><a title="whether a Weil divisor is globally principal" href="_is__Principal.html">isPrincipal</a> -- whether a Weil divisor is globally principal</span></span></li> <li><span><kbd>isQCartier(ZZ,QWeilDivisor)</kbd> -- see <span><a title="whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1." href="_is__Q__Cartier.html">isQCartier</a> -- whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.</span></span></li> <li><span><kbd>isQCartier(ZZ,WeilDivisor)</kbd> -- see <span><a title="whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1." href="_is__Q__Cartier.html">isQCartier</a> -- whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.</span></span></li> <li><span><kbd>isQLinearEquivalent(ZZ,QWeilDivisor,QWeilDivisor)</kbd> -- see <span><a title="whether two Q-divisors are linearly equivalent" href="_is__Q__Linear__Equivalent.html">isQLinearEquivalent</a> -- whether two Q-divisors are linearly equivalent</span></span></li> <li><span><kbd>isReduced(BasicDivisor)</kbd> -- see <span><a title="whether a divisor is reduced" href="_is__Reduced.html">isReduced</a> -- whether a divisor is reduced</span></span></li> <li><span><a title="whether R mod the ideal is smooth" href="_is__Smooth_lp__Ideal_rp.html">isSmooth(Ideal)</a> -- whether R mod the ideal is smooth</span></li> <li><span><kbd>isSNC(BasicDivisor)</kbd> -- see <span><a title="whether the divisor is simple normal crossings" href="_is__S__N__C.html">isSNC</a> -- whether the divisor is simple normal crossings</span></span></li> <li><span><a title="whether a divisor is very ample." href="_is__Very__Ample_lp__Weil__Divisor_rp.html">isVeryAmple(WeilDivisor)</a> -- whether a divisor is very ample.</span></li> <li><span><kbd>isWeilDivisor(RWeilDivisor)</kbd> -- see <span><a title="whether a rational/real divisor is in actuality a Weil divisor" href="_is__Weil__Divisor.html">isWeilDivisor</a> -- whether a rational/real divisor is in actuality a Weil divisor</span></span></li> <li><span><a title="whether a divisor is valid" href="_is__Well__Defined_lp__Basic__Divisor_rp.html">isWellDefined(BasicDivisor)</a> -- whether a divisor is valid</span></li> <li><span><kbd>isZeroDivisor(BasicDivisor)</kbd> -- see <span><a title="whether the divisor is the zero divisor" href="_is__Zero__Divisor.html">isZeroDivisor</a> -- whether the divisor is the zero divisor</span></span></li> <li><span><kbd>mapToProjectiveSpace(WeilDivisor)</kbd> -- see <span><a title="compute the map to projective space associated with the global sections of a Cartier divisor" href="_map__To__Projective__Space.html">mapToProjectiveSpace</a> -- compute the map to projective space associated with the global sections of a Cartier divisor</span></span></li> <li><span><span class="tt">net(BasicDivisor)</span> (missing documentation)<!--tag: (net,BasicDivisor)--> </span></li> <li><span><kbd>nonCartierLocus(WeilDivisor)</kbd> -- see <span><a title="the non-Cartier locus of a Weil divisor" href="_non__Cartier__Locus.html">nonCartierLocus</a> -- the non-Cartier locus of a Weil divisor</span></span></li> <li><span><a title="multiply a divisor by a number" href="___Number_sp_st_sp__Basic__Divisor.html">Number * BasicDivisor</a> -- multiply a divisor by a number</span></li> <li><span><kbd>QQ * RWeilDivisor</kbd> -- see <span><a title="multiply a divisor by a number" href="___Number_sp_st_sp__Basic__Divisor.html">Number * BasicDivisor</a> -- multiply a divisor by a number</span></span></li> <li><span><kbd>QQ * WeilDivisor</kbd> -- see <span><a title="multiply a divisor by a number" href="___Number_sp_st_sp__Basic__Divisor.html">Number * BasicDivisor</a> -- multiply a divisor by a number</span></span></li> <li><span><kbd>RR * QWeilDivisor</kbd> -- see <span><a title="multiply a divisor by a number" href="___Number_sp_st_sp__Basic__Divisor.html">Number * BasicDivisor</a> -- multiply a divisor by a number</span></span></li> <li><span><kbd>RR * RWeilDivisor</kbd> -- see <span><a title="multiply a divisor by a number" href="___Number_sp_st_sp__Basic__Divisor.html">Number * BasicDivisor</a> -- multiply a divisor by a number</span></span></li> <li><span><kbd>negativePart(RWeilDivisor)</kbd> -- see <span><a title="get the effective part or anti-effective part of a divisor" href="_positive__Part.html">positivePart</a> -- get the effective part or anti-effective part of a divisor</span></span></li> <li><span><kbd>positivePart(RWeilDivisor)</kbd> -- see <span><a title="get the effective part or anti-effective part of a divisor" href="_positive__Part.html">positivePart</a> -- get the effective part or anti-effective part of a divisor</span></span></li> <li><span><kbd>primes(BasicDivisor)</kbd> -- see <span><a title="get the list of height-one primes in the support of a divisor" href="_primes.html">primes</a> -- get the list of height-one primes in the support of a divisor</span></span></li> <li><span><a title="pullback a divisor under a ring map" href="_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html">pullback(RingMap,RWeilDivisor)</a> -- pullback a divisor under a ring map</span></li> <li><span><kbd>ramificationDivisor(RingMap)</kbd> -- see <span><a title="compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base" href="_ramification__Divisor.html">ramificationDivisor</a> -- compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base</span></span></li> <li><span><kbd>reflexify(Ideal)</kbd> -- see <span><a title="calculate the double dual of an ideal or module Hom(Hom(M, R), R)" href="_reflexify.html">reflexify</a> -- calculate the double dual of an ideal or module Hom(Hom(M, R), R)</span></span></li> <li><span><kbd>reflexify(Module)</kbd> -- see <span><a title="calculate the double dual of an ideal or module Hom(Hom(M, R), R)" href="_reflexify.html">reflexify</a> -- calculate the double dual of an ideal or module Hom(Hom(M, R), R)</span></span></li> <li><span><kbd>reflexivePower(ZZ,Ideal)</kbd> -- see <span><a title="computes a reflexive power of an ideal in a normal domain" href="_reflexive__Power.html">reflexivePower</a> -- computes a reflexive power of an ideal in a normal domain</span></span></li> <li><span><a title="get the ambient ring of a divisor" href="_ring_lp__Basic__Divisor_rp.html">ring(BasicDivisor)</a> -- get the ambient ring of a divisor</span></li> <li><span><a title="whether two divisors are equal" href="___R__Weil__Divisor_sp_eq_eq_sp__R__Weil__Divisor.html">RWeilDivisor == RWeilDivisor</a> -- whether two divisors are equal</span></li> <li><span><kbd>toQWeilDivisor(QWeilDivisor)</kbd> -- see <span><a title="create a Q-Weil divisor from a Weil divisor" href="_to__Q__Weil__Divisor.html">toQWeilDivisor</a> -- create a Q-Weil divisor from a Weil divisor</span></span></li> <li><span><kbd>toQWeilDivisor(WeilDivisor)</kbd> -- see <span><a title="create a Q-Weil divisor from a Weil divisor" href="_to__Q__Weil__Divisor.html">toQWeilDivisor</a> -- create a Q-Weil divisor from a Weil divisor</span></span></li> <li><span><kbd>torsionSubmodule(Module)</kbd> -- see <span><a title="create the torsion submodule of a module" href="_torsion__Submodule.html">torsionSubmodule</a> -- create the torsion submodule of a module</span></span></li> <li><span><kbd>toRWeilDivisor(QWeilDivisor)</kbd> -- see <span><a title="create a R-divisor from a Q or Weil divisor" href="_to__R__Weil__Divisor.html">toRWeilDivisor</a> -- create a R-divisor from a Q or Weil divisor</span></span></li> <li><span><kbd>toRWeilDivisor(RWeilDivisor)</kbd> -- see <span><a title="create a R-divisor from a Q or Weil divisor" href="_to__R__Weil__Divisor.html">toRWeilDivisor</a> -- create a R-divisor from a Q or Weil divisor</span></span></li> <li><span><kbd>toRWeilDivisor(WeilDivisor)</kbd> -- see <span><a title="create a R-divisor from a Q or Weil divisor" href="_to__R__Weil__Divisor.html">toRWeilDivisor</a> -- create a R-divisor from a Q or Weil divisor</span></span></li> <li><span><kbd>toWeilDivisor(RWeilDivisor)</kbd> -- see <span><a title="create a Weil divisor from a Q or R-divisor" href="_to__Weil__Divisor.html">toWeilDivisor</a> -- create a Weil divisor from a Q or R-divisor</span></span></li> <li><span><a title="trims the ideals displayed to the user and removes primes with coefficient zero" href="_trim_lp__Basic__Divisor_rp.html">trim(BasicDivisor)</a> -- trims the ideals displayed to the user and removes primes with coefficient zero</span></li> <li><span><kbd>zeroDivisor(Ring)</kbd> -- see <span><a title="constructs the zero Weil divisor for the ring" href="_zero__Divisor.html">zeroDivisor</a> -- constructs the zero Weil divisor for the ring</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="an option used to tell divisor construction that a particular ambient ring is expected." href="___Ambient__Ring.html">AmbientRing</a> -- an option used to tell divisor construction that a particular ambient ring is expected.</span></li> <li><span><a title="an option used to tell divisor construction that a particular type of coefficients are expected." href="___Coefficient__Type.html">CoefficientType</a> -- an option used to tell divisor construction that a particular type of coefficients are expected.</span></li> <li><span><a title="a symbol used as a key within the divisor cache" href="_ideals.html">ideals</a> -- a symbol used as a key within the divisor cache</span></li> <li><span><a title="a valid value for the Strategy option in dualize or reflexify" href="___Ideal__Strategy.html">IdealStrategy</a> -- a valid value for the Strategy option in dualize or reflexify</span></li> <li><span><kbd>ModuleStrategy</kbd> -- see <span><a title="a valid value for the Strategy option in dualize or reflexify" href="___Ideal__Strategy.html">IdealStrategy</a> -- a valid value for the Strategy option in dualize or reflexify</span></span></li> <li><span><kbd>NoStrategy</kbd> -- see <span><a title="a valid value for the Strategy option in dualize or reflexify" href="___Ideal__Strategy.html">IdealStrategy</a> -- a valid value for the Strategy option in dualize or reflexify</span></span></li> <li><span><a title="an option used by numerous functions which tells it to treat the divisors as if we were working on the Proj of the ambient ring." href="___Is__Graded.html">IsGraded</a> -- an option used by numerous functions which tells it to treat the divisors as if we were working on the Proj of the ambient ring.</span></li> <li><span><a title="an option used to specify to certain functions that we know that the divisor is Cartier" href="___Known__Cartier.html">KnownCartier</a> -- an option used to specify to certain functions that we know that the divisor is Cartier</span></li> <li><span><a title="an option used to specify to certain functions that we know that the ring is a domain" href="___Known__Domain.html">KnownDomain</a> -- an option used to specify to certain functions that we know that the ring is a domain</span></li> <li><span><a title="an option used by embedAsIdeal how many times to try embedding the module as an ideal in a random way." href="___M__Tries.html">MTries</a> -- an option used by embedAsIdeal how many times to try embedding the module as an ideal in a random way.</span></li> <li><span><a title="a value for the option Strategy for the pullback method" href="___Primes.html">Primes</a> -- a value for the option Strategy for the pullback method</span></li> <li><span><a title="an option for embedAsIdeal" href="___Return__Map.html">ReturnMap</a> -- an option for embedAsIdeal</span></li> <li><span><a title="an option used to tell functions whether not to do checks." href="___Safe.html">Safe</a> -- an option used to tell functions whether not to do checks.</span></li> <li><span><a title="an option used in a number of functions" href="___Section.html">Section</a> -- an option used in a number of functions</span></li> <li><span><a title="a value for the option Strategy for the pullback method" href="___Sheaves.html">Sheaves</a> -- a value for the option Strategy for the pullback method</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="divisors" href="index.html">Divisor</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Divisor.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Divisor.m2:1966:0</span>.</p> </div> </div> </div> </body> </html>