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Given a variety $X=\{F:=x_0^3+t(x_1^3+x_2^3+x_3^3+x_4^3+1)=0\}\subset \mathbb{C}^6$, where $(x_0,...,x_4,t)$ are coordinates of $\mathbb{C}^6$. How to resolve the singularity by only blow-up smooth subvariety of $X$ that is contained in $\{x_0^3=0\}$? How many steps do we need for blow-ups, and what are the exceptional divisors for each step?

Can we even use computers? :-)

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This might not be a complete answer, but it's far too long for a comment.

The software package singular might help: It can resolve singularities. It can resolve yours as well, quite fast actually.

I used this code:

LIB"resolve.lib";
ring R=0,(a,b,c,d,e,t),dp;
ideal I=a^3+t*(b^3+c^3+d^3+e^3+1);
list L=resolve(I,1);

The complete output is as follows:

                     SINGULAR                                 /
 A Computer Algebra System for Polynomial Computations       /   version 3-1-6
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Dec 2012
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
> LIB"resolve.lib";
// ** loaded /usr/share/Singular/LIB/resolve.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/sing.lib (14840,2012-04-20)
// ** loaded /usr/share/Singular/LIB/random.lib (14661,2012-03-05)
// ** loaded /usr/share/Singular/LIB/matrix.lib (13658,2010-11-16)
// ** loaded /usr/share/Singular/LIB/nctools.lib (14246,2011-05-26)
// ** loaded /usr/share/Singular/LIB/poly.lib (14852,2012-04-30)
// ** loaded /usr/share/Singular/LIB/ring.lib (15322,2012-10-12)
// ** loaded /usr/share/Singular/LIB/general.lib (14191,2011-05-04)
// ** loaded /usr/share/Singular/LIB/inout.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/linalg.lib (13733,2010-12-06)
// ** loaded /usr/share/Singular/LIB/presolve.lib (14203,2011-05-05)
// ** loaded /usr/share/Singular/LIB/primdec.lib (14732,2012-03-30)
// ** loaded /usr/share/Singular/LIB/absfact.lib (14191,2011-05-04)
// ** loaded /usr/share/Singular/LIB/triang.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/elim.lib (14661,2012-03-05)
> ring R=0,(a,b,c,d,e,t),dp;
> ideal I=a^3+t*(b^3+c^3+d^3+e^3+1);
> list L=resolve(I,1);
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 0
Total number of charts currently in chart-tree: 1
Index of the current chart in chart-tree: 1
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=0

==== Ideal of Variety:
_[1]=b3t+c3t+d3t+e3t+a3+t

==== Exceptional Divisors:
empty list

==== Images of variables of original ring:
_[1]=a
_[2]=b
_[3]=c
_[4]=d
_[5]=e
_[6]=t

-------------------------- Upcoming Center ---------------------
_[1]=t
_[2]=a
_[3]=b3+c3+d3+e3+1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 0
Total number of charts currently in chart-tree: 4
Index of the current chart in chart-tree: 2
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(6)*y(2)+1

==== Ideal of Variety:
_[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(6)*y(2)+1
_[2]=x(6)*y(1)^3+y(2)

==== Exceptional Divisors:
[1]:
   _[1]=x(6)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1

==== Images of variables of original ring:
_[1]=x(6)*y(1)
_[2]=x(2)
_[3]=x(3)
_[4]=x(4)
_[5]=x(5)
_[6]=x(6)

-------------------------- Upcoming Center ---------------------
_[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(6)*y(2)+1
_[2]=x(6)*y(1)^3+y(2)
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 1
Total number of charts currently in chart-tree: 4
Index of the current chart in chart-tree: 3
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(1)*y(2)+1

==== Ideal of Variety:
_[1]=y(0)*y(2)+x(1)
_[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(1)*y(2)+1

==== Exceptional Divisors:
[1]:
   _[1]=x(1)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1

==== Images of variables of original ring:
_[1]=x(1)
_[2]=x(2)
_[3]=x(3)
_[4]=x(4)
_[5]=x(5)
_[6]=x(1)*y(0)

-------------------------- Upcoming Center ---------------------
_[1]=y(2)
_[2]=y(0)
_[3]=x(1)
_[4]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 1
Total number of charts currently in chart-tree: 6
Index of the current chart in chart-tree: 4
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=0

==== Ideal of Variety:
_[1]=x(2)^3*y(1)^3+x(3)^3*y(1)^3+x(4)^3*y(1)^3+x(5)^3*y(1)^3+y(1)^3+y(0)

==== Exceptional Divisors:
[1]:
   _[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1

==== Images of variables of original ring:
_[1]=x(2)^3*y(1)+x(3)^3*y(1)+x(4)^3*y(1)+x(5)^3*y(1)+y(1)
_[2]=x(2)
_[3]=x(3)
_[4]=x(4)
_[5]=x(5)
_[6]=x(2)^3*y(0)+x(3)^3*y(0)+x(4)^3*y(0)+x(5)^3*y(0)+y(0)

-------------------------- Upcoming Center ---------------------
_[1]=x(2)^3*y(1)^3+x(3)^3*y(1)^3+x(4)^3*y(1)^3+x(5)^3*y(1)^3+y(1)^3+y(0)
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 2
Total number of charts currently in chart-tree: 6
Index of the current chart in chart-tree: 5
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(7)^2*y(2)+1

==== Ideal of Variety:
_[1]=x(7)*y(1)+y(2)
_[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3-x(7)^2*y(2)+1

==== Exceptional Divisors:
[1]:
   _[1]=y(2)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1
[2]:
   _[1]=x(7)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1

==== Images of variables of original ring:
_[1]=x(7)*y(2)
_[2]=x(2)
_[3]=x(3)
_[4]=x(4)
_[5]=x(5)
_[6]=x(7)^2*y(1)*y(2)

-------------------------- Upcoming Center ---------------------
_[1]=y(2)
_[2]=x(7)
_[3]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 2
Total number of charts currently in chart-tree: 8
Index of the current chart in chart-tree: 6
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(6)^2*y(0)*y(2)-x(2)^3-x(3)^3-x(4)^3-x(5)^3-1

==== Ideal of Variety:
_[1]=x(6)*y(0)+y(2)
_[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+x(6)*y(2)^2+1

==== Exceptional Divisors:
[1]:
   _[1]=y(2)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1
[2]:
   _[1]=x(6)
   _[2]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1

==== Images of variables of original ring:
_[1]=x(6)*y(2)
_[2]=x(2)
_[3]=x(3)
_[4]=x(4)
_[5]=x(5)
_[6]=x(6)^2*y(2)

-------------------------- Upcoming Center ---------------------
_[1]=y(2)
_[2]=x(6)
_[3]=x(2)^3+x(3)^3+x(4)^3+x(5)^3+1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 2
Total number of charts currently in chart-tree: 10
Index of the current chart in chart-tree: 7
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(7)^3*y(1)^2-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Ideal of Variety:
_[1]=x(6)*y(1)+1
_[2]=x(1)^3*x(6)+x(2)^3*x(6)+x(3)^3*x(6)+x(4)^3*x(6)+x(7)^3*y(1)+x(6)
_[3]=x(7)^3*y(1)^2-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Exceptional Divisors:
[1]:
   _[1]=1
[2]:
   _[1]=y(1)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
[3]:
   _[1]=x(7)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1

==== Images of variables of original ring:
_[1]=x(7)^2*y(1)
_[2]=x(1)
_[3]=x(2)
_[4]=x(3)
_[5]=x(4)
_[6]=x(6)*x(7)^3*y(1)^2

-------------------------- Upcoming Center ---------------------
_[1]=x(6)*y(1)+1
_[2]=x(1)^3*x(6)+x(2)^3*x(6)+x(3)^3*x(6)+x(4)^3*x(6)+x(7)^3*y(1)+x(6)
_[3]=x(7)^3*y(1)^2-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 3
Total number of charts currently in chart-tree: 10
Index of the current chart in chart-tree: 8
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(5)^3*y(0)-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Ideal of Variety:
_[1]=x(6)+y(0)
_[2]=x(5)^3*y(0)-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Exceptional Divisors:
[1]:
   _[1]=y(0)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
[2]:
   _[1]=1
[3]:
   _[1]=x(5)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1

==== Images of variables of original ring:
_[1]=x(5)^2*y(0)
_[2]=x(1)
_[3]=x(2)
_[4]=x(3)
_[5]=x(4)
_[6]=x(5)^3*x(6)*y(0)

-------------------------- Upcoming Center ---------------------
_[1]=x(6)+y(0)
_[2]=x(5)^3*y(0)-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 4
Total number of charts currently in chart-tree: 10
Index of the current chart in chart-tree: 9
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(6)*x(7)^3*y(1)^2-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Ideal of Variety:
_[1]=x(6)*y(1)+1
_[2]=x(7)^3*y(1)+x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
_[3]=x(1)^3*x(6)+x(2)^3*x(6)+x(3)^3*x(6)+x(4)^3*x(6)-x(7)^3+x(6)

==== Exceptional Divisors:
[1]:
   _[1]=1
[2]:
   _[1]=y(1)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
[3]:
   _[1]=x(7)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1

==== Images of variables of original ring:
_[1]=x(7)^2*y(1)
_[2]=x(1)
_[3]=x(2)
_[4]=x(3)
_[5]=x(4)
_[6]=x(7)^3*y(1)^2

-------------------------- Upcoming Center ---------------------
_[1]=x(6)*y(1)+1
_[2]=x(7)^3*y(1)+x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
_[3]=x(1)^3*x(6)+x(2)^3*x(6)+x(3)^3*x(6)+x(4)^3*x(6)-x(7)^3+x(6)
----------------------------------------------------------------
++++++++++++++ Overview of Current Chart +++++++++++++++++++++++
Current number of final charts: 5
Total number of charts currently in chart-tree: 10
Index of the current chart in chart-tree: 10
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

==== Ambient Space:
_[1]=x(5)^3*x(6)*y(0)-x(1)^3-x(2)^3-x(3)^3-x(4)^3-1

==== Ideal of Variety:
_[1]=x(6)+y(0)
_[2]=x(5)^3*y(0)^2+x(1)^3+x(2)^3+x(3)^3+x(4)^3+1

==== Exceptional Divisors:
[1]:
   _[1]=y(0)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
[2]:
   _[1]=1
[3]:
   _[1]=x(5)
   _[2]=x(1)^3+x(2)^3+x(3)^3+x(4)^3+1

==== Images of variables of original ring:
_[1]=x(5)^2*y(0)
_[2]=x(1)
_[3]=x(2)
_[4]=x(3)
_[5]=x(4)
_[6]=x(5)^3*y(0)

-------------------------- Upcoming Center ---------------------
_[1]=x(6)+y(0)
_[2]=x(5)^3*y(0)^2+x(1)^3+x(2)^3+x(3)^3+x(4)^3+1
----------------------------------------------------------------
============= result will be tested ==========

the number of charts obtained: 6
=============     result is o.k.    ==========

However, I am too unexperienced with singular to be sure how to interpret the output exactly. You might have to read this tutorial quite carefully:

http://www.singular.uni-kl.de/tutor_resol.ps

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  • $\begingroup$ Thank you for your answer. It is very powerful. However, I am worrying about if the procedure of blowing-up just blow up smooth sub variety from the original singular set. Does it change the original smooth locus? $\endgroup$ – Feng Hao Apr 17 '16 at 2:08

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