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GlicciPointsInP3 :: tangentDimension

tangentDimension -- compute the dimension of tangentspaces

Synopsis

Description

J ⊂ I is a Gorenstein ideal J of distinct points contained in the ideal I of a subset set. i.e. the situation of Gorenstein linkage. The degree of 0 pieces of the Hom spaces above have the following interpretation: Hom(J,S/J)0 and Hom(I,S/I)0 are the spaces of first order deformations with constant Hilbert function (not only constant Hilbert polynomial!) Hom(I/J,S/J)0 ⊂ Hom(J,S/J)0 are those defomations of J, which do not move I, i.e the tangent space of the fiber from HCXcG -> HCX at the point (J,I). Example:
i1 : S=ZZ/10009[a..d];hV={1,3,3,1},n=3;
i3 : time (J,I,M)=findSplittingExample(S,hV,n);
     -- used 0.0470565 seconds
i4 : hV = hVector degree J

o4 = {1, 3, 4}

o4 : List
i5 : betti res J

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

o5 : BettiTally
i6 : betti res I

            0 1 2 3
o6 = total: 1 4 5 2
         0: 1 1 . .
         1: . 3 5 2

o6 : BettiTally
i7 : time (tf,tg,th)=tangentDimension(I,J)
     -- used 0.00998469 seconds

o7 = (12, 21, 9)

o7 : Sequence
i8 : tf==tg-th

o8 = true
i9 : I'=J:I;

o9 : Ideal of S
i10 : time tangentDimension(I,I',J)
     -- used 0.00820033 seconds

o10 = ((12, 21, 9), (6, 21, 15))

o10 : Sequence
i11 : degree I, degree J, degree I'

o11 = (3, 8, 5)

o11 : Sequence
It follows that the map HCXuY=G -> Hilb3xHilb5 is a bi-dominant correspondence.

Ways to use tangentDimension :