Using Macaulay 2 to find free divisors.












3












$begingroup$


Given a hypersurface $D = h^{-1}(0)$ for some polynomial $h in mathbb{C} [x,y,z]$ I want to be able to use Macaulay 2 to tell if it's a free divisor or not.



What I've got so far;



Let $h_{p}$ be the reduced equation for $h$. For $D$ to be a free divisor I need $Der(-logD) := { delta : delta(h_{p}) in (h_{p})}$ to be a locally free $mathcal{O}_{mathbb{C} ^{n}}$-module, where $delta$ is a logarithmic vector field.
So for a vector field $delta = a frac{partial}{partial{x}} + b frac{partial}{partial{y}} + c frac{partial}{partial{z}}$ I need $a frac{partial{h}}{partial{x}} + b frac{partial{h}}{partial{y}} + c frac{partial{h}}{partial{z}} = r h_{p}$ for some $r$.



So I think If I remove the bottom row of ones that I get when when using Macaulay 2 to calculate the kernel of the map given by the matrix $lbrack frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h_{p} rbrack$ I'll get the module $Der(-logD)$.



But I'm not sure how to then check if its a free module, I am slightly confused too as I've been told to use the resolution function in Macaulay 2 but when ever I try I can't seem to get it to work.



I've only just started using Macaulay 2 and I am fairly new to free divisors too so sorry for any errors and thanks in advance for any help.






algebraic-geometry math-software macaulay2







share|cite|improve this question











$endgroup$












  • $begingroup$
    Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 9:34










  • $begingroup$
    When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:40












  • $begingroup$
    $= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:56










  • $begingroup$
    Sorry for the formatting It was the best I could do.
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:57










  • $begingroup$
    You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 16:40
















3












$begingroup$


Given a hypersurface $D = h^{-1}(0)$ for some polynomial $h in mathbb{C} [x,y,z]$ I want to be able to use Macaulay 2 to tell if it's a free divisor or not.



What I've got so far;



Let $h_{p}$ be the reduced equation for $h$. For $D$ to be a free divisor I need $Der(-logD) := { delta : delta(h_{p}) in (h_{p})}$ to be a locally free $mathcal{O}_{mathbb{C} ^{n}}$-module, where $delta$ is a logarithmic vector field.
So for a vector field $delta = a frac{partial}{partial{x}} + b frac{partial}{partial{y}} + c frac{partial}{partial{z}}$ I need $a frac{partial{h}}{partial{x}} + b frac{partial{h}}{partial{y}} + c frac{partial{h}}{partial{z}} = r h_{p}$ for some $r$.



So I think If I remove the bottom row of ones that I get when when using Macaulay 2 to calculate the kernel of the map given by the matrix $lbrack frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h_{p} rbrack$ I'll get the module $Der(-logD)$.



But I'm not sure how to then check if its a free module, I am slightly confused too as I've been told to use the resolution function in Macaulay 2 but when ever I try I can't seem to get it to work.



I've only just started using Macaulay 2 and I am fairly new to free divisors too so sorry for any errors and thanks in advance for any help.






algebraic-geometry math-software macaulay2







share|cite|improve this question











$endgroup$












  • $begingroup$
    Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 9:34










  • $begingroup$
    When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:40












  • $begingroup$
    $= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:56










  • $begingroup$
    Sorry for the formatting It was the best I could do.
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:57










  • $begingroup$
    You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 16:40














3












3








3





$begingroup$


Given a hypersurface $D = h^{-1}(0)$ for some polynomial $h in mathbb{C} [x,y,z]$ I want to be able to use Macaulay 2 to tell if it's a free divisor or not.



What I've got so far;



Let $h_{p}$ be the reduced equation for $h$. For $D$ to be a free divisor I need $Der(-logD) := { delta : delta(h_{p}) in (h_{p})}$ to be a locally free $mathcal{O}_{mathbb{C} ^{n}}$-module, where $delta$ is a logarithmic vector field.
So for a vector field $delta = a frac{partial}{partial{x}} + b frac{partial}{partial{y}} + c frac{partial}{partial{z}}$ I need $a frac{partial{h}}{partial{x}} + b frac{partial{h}}{partial{y}} + c frac{partial{h}}{partial{z}} = r h_{p}$ for some $r$.



So I think If I remove the bottom row of ones that I get when when using Macaulay 2 to calculate the kernel of the map given by the matrix $lbrack frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h_{p} rbrack$ I'll get the module $Der(-logD)$.



But I'm not sure how to then check if its a free module, I am slightly confused too as I've been told to use the resolution function in Macaulay 2 but when ever I try I can't seem to get it to work.



I've only just started using Macaulay 2 and I am fairly new to free divisors too so sorry for any errors and thanks in advance for any help.






algebraic-geometry math-software macaulay2







share|cite|improve this question











$endgroup$




Given a hypersurface $D = h^{-1}(0)$ for some polynomial $h in mathbb{C} [x,y,z]$ I want to be able to use Macaulay 2 to tell if it's a free divisor or not.



What I've got so far;



Let $h_{p}$ be the reduced equation for $h$. For $D$ to be a free divisor I need $Der(-logD) := { delta : delta(h_{p}) in (h_{p})}$ to be a locally free $mathcal{O}_{mathbb{C} ^{n}}$-module, where $delta$ is a logarithmic vector field.
So for a vector field $delta = a frac{partial}{partial{x}} + b frac{partial}{partial{y}} + c frac{partial}{partial{z}}$ I need $a frac{partial{h}}{partial{x}} + b frac{partial{h}}{partial{y}} + c frac{partial{h}}{partial{z}} = r h_{p}$ for some $r$.



So I think If I remove the bottom row of ones that I get when when using Macaulay 2 to calculate the kernel of the map given by the matrix $lbrack frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h_{p} rbrack$ I'll get the module $Der(-logD)$.



But I'm not sure how to then check if its a free module, I am slightly confused too as I've been told to use the resolution function in Macaulay 2 but when ever I try I can't seem to get it to work.



I've only just started using Macaulay 2 and I am fairly new to free divisors too so sorry for any errors and thanks in advance for any help.






algebraic-geometry math-software macaulay2




algebraic-geometry math-software macaulay2






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 9:33









Rodrigo de Azevedo

12.9k41856




12.9k41856










asked Oct 22 '13 at 17:44









Geraint JonesGeraint Jones

234




234












  • $begingroup$
    Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 9:34










  • $begingroup$
    When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:40












  • $begingroup$
    $= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:56










  • $begingroup$
    Sorry for the formatting It was the best I could do.
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:57










  • $begingroup$
    You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 16:40


















  • $begingroup$
    Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 9:34










  • $begingroup$
    When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:40












  • $begingroup$
    $= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:56










  • $begingroup$
    Sorry for the formatting It was the best I could do.
    $endgroup$
    – Geraint Jones
    Oct 23 '13 at 15:57










  • $begingroup$
    You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
    $endgroup$
    – Fredrik Meyer
    Oct 23 '13 at 16:40
















$begingroup$
Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
$endgroup$
– Fredrik Meyer
Oct 23 '13 at 9:34




$begingroup$
Well, a free module have no relations among its generators. What happens when you try to make a resolution of $Der(-log D)$?
$endgroup$
– Fredrik Meyer
Oct 23 '13 at 9:34












$begingroup$
When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:40






$begingroup$
When I calculate the kernel of the matrix $[frac{partial{h}}{partial{x}} , frac{partial{h}}{partial{y}} , frac{partial{h}}{partial{z}} , -h ]$ I get the message (in this case I'm using h=xyz): -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) and then I get $Der(-logD)$ by getting the generators of this then by rewriting it without the bottom row of ones and calculating its image. Then when I use resolution on this I get, (in next comment)
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:40














$begingroup$
$= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:56




$begingroup$
$= 0 : R^{3} <--------------------- R^{3} : 0$ then on the next row $begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z end{bmatrix} $ then the next row is $1 : <---- : 1$ and the final row is just $0$
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:56












$begingroup$
Sorry for the formatting It was the best I could do.
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:57




$begingroup$
Sorry for the formatting It was the best I could do.
$endgroup$
– Geraint Jones
Oct 23 '13 at 15:57












$begingroup$
You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
$endgroup$
– Fredrik Meyer
Oct 23 '13 at 16:40




$begingroup$
You get the error message because you're working over $mathbb C$, try using R=QQ[x,y,z] instead. Let K=ker M. Then write "prune K".
$endgroup$
– Fredrik Meyer
Oct 23 '13 at 16:40










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