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D.6.4.2 CurveRes
Procedure from library curvepar.lib (see curvepar_lib).
- Usage:
- CurveRes(I);
I ideal
- Assume:
- The basering is r=0,(x(1..n))
V(I) is a curve with a singular point 0.
- Compute:
- Resolution of the curve V(I).
- Return:
- a ring R=basering+k[a]
Ring R contains a list Resolve
Resolve is a list of charts
Each Resolve[i] is a list of size 6
Resolve[i][1] is an ideal J of a new curve. J=teta(I).
Resolve[i][2] ideal which represents the map
teta:k[x(1)..x(n),a]-->k[x(1)..x(n),a] from the
new curve to the old one.
Resolve[i][3] is an irreducible poly g in k[a]. It is a minimal polynomial for the
new parameter a. deg(g) gives the number of branches in Resolve[i]
Resolve[i][4] sequence of multiplicities (sum over all branches in Resolve as long as
they intersect each other !)
Resolve[i][5] is a list of integers l. It shows, which standard charts we considered.
Resolve[i][6] HN matrix
Resolve[i][7] (only for plane curves) the development of exceptional divisors
the entries correspond to the i-th blowing up. The first entry is an
intvec. The first negative entry gives the splitting of the (over Q
irreducible) branches. The second entry is a list of the exceptional
divisors. If the entry is an integer i, it says that the divisor is not
visible in this chart after the i-th blowing up.
Example:
| | LIB "curvepar.lib";
ring r=0,(x,y,z),dp;
ideal i=x2-y3,z2-y5;
def s=CurveRes(i);
setring s;
Resolve;
==> [1]:
==> [1]:
==> _[1]=x(1)
==> _[2]=x(1)*x(2)-x(3)^2+2*x(3)
==> [2]:
==> _[1]=x(1)^2*x(2)^5+2*x(1)*x(2)^4+x(2)^3
==> _[2]=x(1)*x(2)^3+x(2)^2
==> _[3]=x(1)^2*x(2)^7*x(3)-x(1)^2*x(2)^7+2*x(1)*x(2)^6*x(3)-2*x(1)*x(2\
)^6+x(2)^5*x(3)-x(2)^5
==> _[4]=a
==> [3]:
==> a
==> [4]:
==> [1]:
==> 4
==> [2]:
==> 2
==> [3]:
==> 2
==> [4]:
==> 2
==> [5]:
==> [1]:
==> 2
==> [2]:
==> 1
==> [3]:
==> 2
==> [4]:
==> 2
==> [6]:
==> [1]:
==> _[1]=0
==> _[2]=1
==> _[3]=0
==> [2]:
==> _[1]=1
==> _[2]=0
==> _[3]=0
==> [3]:
==> _[1]=1
==> _[2]=1
==> _[3]=0
==> [4]:
==> _[1]=0
==> _[2]=1
==> _[3]=-1
==> [2]:
==> [1]:
==> _[1]=x(1)
==> _[2]=x(1)*x(2)-x(3)^2-2*x(3)
==> [2]:
==> _[1]=x(1)^2*x(2)^5+2*x(1)*x(2)^4+x(2)^3
==> _[2]=x(1)*x(2)^3+x(2)^2
==> _[3]=x(1)^2*x(2)^7*x(3)+x(1)^2*x(2)^7+2*x(1)*x(2)^6*x(3)+2*x(1)*x(2\
)^6+x(2)^5*x(3)+x(2)^5
==> _[4]=a
==> [3]:
==> a
==> [4]:
==> [1]:
==> 4
==> [2]:
==> 2
==> [3]:
==> 2
==> [4]:
==> 2
==> [5]:
==> [1]:
==> 2
==> [2]:
==> 1
==> [3]:
==> 2
==> [4]:
==> 2
==> [6]:
==> [1]:
==> _[1]=0
==> _[2]=1
==> _[3]=0
==> [2]:
==> _[1]=1
==> _[2]=0
==> _[3]=0
==> [3]:
==> _[1]=1
==> _[2]=1
==> _[3]=0
==> [4]:
==> _[1]=0
==> _[2]=1
==> _[3]=1
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