DefParam -- deformation parameter

Description

DefParam is the name of an optional argument. Its value is a symbol, which specifies the name of the deformation parameter. Its default value is determined by the loadtime configuration Option DefaultDefParam, which has default value t.

For example, we may use the deformation parameter s:

i1 : S=QQ[x_0..x_4];

i2 : I=minors(2,matrix {{x_0,x_1,x_2,x_3},{x_1,x_2,x_3,x_4}});

o2 : Ideal of S

i3 : F0=gens I

o3 = | -x_1^2+x_0x_2 -x_1x_2+x_0x_3 -x_2^2+x_1x_3 -x_1x_3+x_0x_4
     ------------------------------------------------------------------------
     -x_2x_3+x_1x_4 -x_3^2+x_2x_4 |

             1      6
o3 : Matrix S  <-- S

i4 : (F,R,G,C)=versalDeformation(F0,DefParam=>s);

i5 : sum F

o5 = | x_1s_1+x_0s_2-x_1^2+x_0x_2 -s_1s_3+x_0s_4-x_1x_2+x_0x_3
     ------------------------------------------------------------------------
     -s_2s_3+s_3^2-s_1s_4-x_3s_1-x_2s_2+x_1s_4-x_2^2+x_1x_3
     ------------------------------------------------------------------------
     s_2s_3-s_3^2+x_2s_3-x_1x_3+x_0x_4 -x_4s_1-x_3s_2+x_3s_3-x_2x_3+x_1x_4
     ------------------------------------------------------------------------
     x_4s_3-x_3s_4-x_3^2+x_2x_4 |

                       1                6
o5 : Matrix (S[s ..s ])  <-- (S[s ..s ])
                1   4            1   4

See also

• firstOrderDeformations -- use tangent space to create first order perturbations and find relations
• versalDeformation -- computes a power series representation of a versal deformation
• localHilbertScheme -- computes a power series representation of the local Hilbert scheme

Functions with optional argument named DefParam:

• firstOrderDeformations(...,DefParam=>...) -- see firstOrderDeformations -- use tangent space to create first order perturbations and find relations
• localHilbertScheme(...,DefParam=>...) -- see localHilbertScheme -- computes a power series representation of the local Hilbert scheme
• versalDeformation(...,DefParam=>...) -- see versalDeformation(Matrix) -- computes a power series representation of a versal deformation