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 Post subject: Fitting ideals on a variety
PostPosted: Wed May 15, 2013 6:08 am 

Joined: Thu Mar 25, 2010 2:25 pm
Posts: 5

First of all I am not a great programmer. So I ask your help in following question
usually I have to look at specific polynomial map f and its differential/Jacobian on a
particular variety.

Now what I would like to do is to compute the rank dropping sets of the Jacobian df
in a particular variety V(I), I=<g_1,...,g_n>\subset K[x_1,...,x_k]=A.

This of course lead to Fitting ideals which has to vanish for a sequence of coranks of df on a variety V
and to the 1st syzygy module of dg, g:=(g_1,...,g_n).

The problem that the variety V(g_1,...,g_n) might have a fairly big dimension let's say l and in order to compute the first rank dropping set we should look at the Fitting ideal spanned by k-l minors of dg. The number of minors might be huge.

So in order to simplify matters could somebody program a code which would do the following:

Always when a one minor is computed we reduce it by Gröbner basis of I and if it vanishes we can discard it because we are looking at the rank dropping sets on V(I).

Also if we have a completely different map f:=(f_1,..,f_s) and its Jacobian df
and we want to look at the rank dropping sets at V(I) so could somebody write a code
or suggest how to do the following:

1) We want to look at all rank dropping sets of df on V
2) When we generate the ideal spanned by the minors we would at every step when a new minor is computed we would automatically test if it vanishes on V and discard and if so discard it.
3) Finally we would end up with relevant polynomials which do not automatically vanish on V and these minors would give us the relevant minors M=<m1,> which would give us the relevant conditions so that the rank of df drops i.e the ''singular'' variety

4) So in a nutshell: I would only like to look at the minors which actually give some extra conditions for the rank of df to drop on V. Of course the minor which vanish at V(I) do not matter at all !

If somebody has the energy or time I would really appreciate if someone would explain to me how to do this or is it already automatically possible to do this in singular ? For example in a quotient ring of an ideal ?

hope to hear from you soon !


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