findSplittingExample -- find an example of a Gorenstein ideal of points with a subset of degree n

Synopsis

Usage:
(J,I,M)=findSplittingExample(S,d1,d2,n);
(J,I,M)=findSplittingExample(S,hV,n)
Inputs:
- S, a ring, homogeneous coordinate ring of P³
- d1, a list
- d2, a list, degrees of the target and source of the Buchsbaum-Eisenbud skew symmetric matrix,or
- hV, a list, h-vector of the desired Gorenstein ideal J
- n, an integer, the degree of the desired subset defined by I
Outputs:
- J, an ideal, Gorenstein ideal of points
- I, an ideal, of a degree n subscheme
- M, a matrix, Buchsbaum-Eisenbud skew symmetric matrix

Description

We search for a pair (J,I) consisting of a reduced Gorenstein ideal J of points and an ideal I of a subset of degree n by choosing random Gorenstein ideals with an appropriate h-vector and checking to see whether they split in such a way that a subset of the components represents a scheme of degree n

i1 : S=ZZ/10009[x_0..x_3]

o1 = S

o1 : PolynomialRing

i2 : d1={7:0},d2={7:-1},n=10

o2 = ({(0, 0, 0, 0, 0, 0, 0)}, {(-1, -1, -1, -1, -1, -1, -1)}, 10)

o2 : Sequence

i3 : time (J,I,M)=findSplittingExample(S,d1,d2,n);
     -- used 0.0514644 seconds

i4 : degree J, degree I

o4 = (14, 10)

o4 : Sequence

i5 : betti res J

            0 1 2 3
o5 = total: 1 7 7 1
         0: 1 . . .
         1: . . . .
         2: . 7 7 .
         3: . . . .
         4: . . . 1

o5 : BettiTally

i6 : betti res I

            0  1  2 3
o6 = total: 1 10 15 6
         0: 1  .  . .
         1: .  .  . .
         2: . 10 15 6

o6 : BettiTally

i7 : betti res (J:I)

            0 1 2 3
o7 = total: 1 6 8 3
         0: 1 . . .
         1: . 6 8 3

o7 : BettiTally

i8 : S=ZZ/10009[x_0..x_3]

o8 = S

o8 : PolynomialRing

i9 : n=7;hV={1,3,3,3,1}

o10 = {1, 3, 3, 3, 1}

o10 : List

i11 : time (J,I,M)=findSplittingExample(S,hV,n);
     -- used 0.0355073 seconds

i12 : degree J, degree I

o12 = (11, 7)

o12 : Sequence

i13 : betti res J

             0 1 2 3
o13 = total: 1 5 5 1
          0: 1 . . .
          1: . 3 2 .
          2: . . . .
          3: . 2 3 .
          4: . . . 1

o13 : BettiTally

i14 : betti res I

             0 1 2 3
o14 = total: 1 6 8 3
          0: 1 . . .
          1: . 3 2 .
          2: . 3 6 3

o14 : BettiTally

Ways to use `findSplittingExample` :

findSplittingExample(Ring,List,List,ZZ)
findSplittingExample(Ring,List,ZZ)

findSplittingExample -- find an example of a Gorenstein ideal of points with a subset of degree n

Synopsis

Description

Ways to use findSplittingExample :

Ways to use `findSplittingExample` :