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VersalDeformations :: localHilbertScheme

localHilbertScheme -- computes a power series representation of the local Hilbert scheme

Synopsis

Description

F0 should be a matrix with homogeneous entries over some polynomial ring with one row.

Each element of the sequence (F,R,G,C) is a list of matrices in increasing powers of t. Their sums satisfy the deformation equation transpose ((sum F)*(sum R))+(sum C)*(sum G)==0 up to powers of t equal to the length of F. Furthermore, F_0=F0, R_0=gens ker F0, C_0=T^2(0,F_0) and F_1 consists of first order perturbations corresponding to normalModule(0,F0). Thus, F and G represent a universal family and local analytic equations for the Hilbert scheme.

Several options are available to control the termination of the calculation. The calculation will terminate at the very latest after reaching order equal to the option HighestOrder. If any single lifting step takes longer than TimeLimit the algorithm will terminate earlier. If PolynomialCheck is set to true, then the algorithm will check if the present solution lifts to infinite order and terminate if this is the case. If SanityCheck is set to true, then the algorithm will check that the present solution really does solve the deformation equation, and terminate if this is not the case. Finally, Verbosity may be used to control the verbosity of the output.

The option SmartLift is also available, which controls whether the algorithm spends extra time trying to find liftings which introduce no new obstructions at the next highest order. The option CorrectionMatrix may be used to control which liftings are considered.

For example, consider a degenerate twisted cubic curve:

i1 : S=QQ[x,y,z,w];
i2 : F0=matrix {{x*z,y*z,z^2,x^3}}

o2 = | xz yz z2 x3 |

             1       4
o2 : Matrix S  <--- S
i3 : (F,R,G,C)=localHilbertScheme(F0);
Calculating first order deformations and obstruction space
Calculating first order relations
Starting lifting
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Solution is polynomial

Local equations for the Hilbert scheme are thus given by

i4 : sum G

o4 = | t_1t_16              |
     | t_9t_16              |
     | t_4t_16              |
     | t_14t_16-1/2t_15t_16 |

                                                                                      4                                                                                1
o4 : Matrix (S[t , t , t , t , t , t , t , t , t , t  , t  , t  , t  , t  , t  , t  ])  <--- (S[t , t , t , t , t , t , t , t , t , t  , t  , t  , t  , t  , t  , t  ])
                1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16             1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

Caveat

The output may not be the local Hilbert scheme if standard comparison theorems do not hold for the ideal generated by FO.

Ways to use localHilbertScheme :