Description
This interface to
deformMCMModule constructs a versal deformation of
M0 in the general case. That is, there is given a map of varieties $\phi:Y \rightarrow{} \Sigma$ and a module
M0 defined over the fibre over 0 of $\phi$. A versal deformation of
M0 is then constructed in the deformation theory so defined. Thus its inputs consist of
M0 and a ring homomorphism $\phi^*$ representing the map $\phi:Y \rightarrow{} \Sigma$. The procedure requires that
M0 be an MCM module over the ring \mathcal{O}_Y/\phi^*(m_\Sigma), where m_\Sigma is the ideal generated by the variables of \mathcal{O}_\Sigma. Otherwise an error will result.
If
M0 is free, then
deformMCMModule returns
(S,M), where
S is the source of
phi and
M is a free module over the ambient ring of
Y, generated in the same degrees as
M0.
i1 : OSigma = QQ[x, Degrees=>{2}];
|
i2 : OY = QQ[y,z,x, Degrees=>{2,3,2}]/(z^2-(y-x)*y^2);
|
i3 : phi = map(OY,OSigma, {x})
o3 = map (OY, OSigma, {x})
o3 : RingMap OY <-- OSigma
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i4 : use OSigma
o4 = OSigma
o4 : PolynomialRing
|
i5 : OX = trim (OY/phi(ideal x))
o5 = OX
o5 : QuotientRing
|
i6 : (OS,M) = deformMCMModule(module ideal (y,z),phi)
o6 = (OS, cokernel {2} | z+xi_1 y2-yx-yxi_2+xxi_2+xi_2^2 |)
{3} | -y-xi_2 -z+xi_1 |
o6 : Sequence
|
i7 : prune OS
QQ[x, xi ..xi ]
1 2
o7 = -----------------
2 2 3
xi + x*xi + xi
1 2 2
o7 : QuotientRing
|
The above example deforms the maximal ideal of the A2 singularity onto the Whitney umbrella $Y$ (whose ring is
OY above, of which the former is a hyperplane section given by the fibre over 0 of the map $Y \rightarrow{} \Sigma$ defined by $\phi$ above. The resulting base space
S (whose ring
OS is part of the output of
deformMCMModule) is the Hilbert scheme of one point on the Whitney umbrella, which is isomorphic to $Y$. The substitution
y => -xi_2, z => xi_1 shows this isomorphism.
The resulting module $M$, when restricted to the fibre product of $Y$ and $S$ over $\Sigma$, is isomorphic to the ideal defining the diagonal embedding of $Y$.