For each pair (d,hV) in L of a degree d and an possible h-vector hV for a Gorenstein collection of g points, we randomly choose a point (J,I) in the correspondence and check whether this point belongs to a correspondence which dominates both Hilbd and Hilb(g-d). The assertion that J consists of g distinct points is verified during the selection of J. We assert that both I and I’=J:I have generic h-vectors for their degree. We then use the function tangentDimension to compute the tangent dimension tf and tf’ of the fibers of the projections HCGhV to H=Hilbd and H’=Hilb(g-d). We then test the numerical equality of Corollary 3.3 of Twenty Points in P^3. Two lists of candidates are returned: those that passed the test and those that failed. The elements of the lists are tuples
((d,hV),(tf,tg,3d),(tf’,tg,3(g-d)), betti M)
where M is the skew symmetric BE-matrix. The function prints some timings and intermediate results to the screen.
i1 : S=ZZ/10009[x_0..x_3];L=createCandidates 1; |
i3 : (Lgood,Lbad)=checkCandidates(L,S);
-- used 0.00613789 seconds
((1, {1, 1}, 1), true)
-- used 0.00636443 seconds
((2, {1, 1, 1}, 1), true)
-- used 0.00642103 seconds
((2, {1, 1, 1, 1}, 2), true)
-- used 0.00611228 seconds
((2, {1, 2, 1}, 2), true)
-- used 0.00661118 seconds
((3, {1, 2, 1}, 1), true)
-- used 0.0072511 seconds
((3, {1, 2, 2, 1}, 3), true)
-- used 0.0256917 seconds
((3, {1, 3, 1}, 2), true)
-- used 0.0137047 seconds
((4, {1, 3, 1}, 1), true)
-- used 0.00933336 seconds
((4, {1, 3, 3, 1}, 4), true)
-- used 0.00807936 seconds
((5, {1, 3, 3, 1}, 3), true)
-- used 0.00769319 seconds
((6, {1, 3, 3, 1}, 2), true)
-- used 0.00765866 seconds
((7, {1, 3, 3, 1}, 1), true)
|
i4 : #L,#Lgood,#Lbad o4 = (12, 12, 0) o4 : Sequence |
i5 : Lgood_11
0 1
o5 = ((7, {1, 3, 3, 1}), (0, 21, 21), (18, 21, 3), total: 3 3)
2: 3 .
3: . 3
o5 : Sequence
|
i6 : LbadCandidates={(7, {1, 3, 3, 3, 1}),
(7, {1, 3, 3, 3, 3, 1}),
(13, {1, 3, 6, 6, 6, 3, 1}),
(14, {1, 3, 6, 6, 6, 3, 1}),
(15, {1, 3, 6, 6, 6, 3, 1}),
(16, {1, 3, 6, 6, 6, 6, 3, 1}),
(17, {1, 3, 6, 7, 7, 6, 3, 1}),
(25, {1, 3, 6, 10, 10, 10, 6, 3, 1}),
(26, {1, 3, 6, 10, 10, 10, 6, 3, 1})};
|
i7 : LfiniteFiber={(7, {1, 3, 3, 1}),
(17, {1, 3, 6, 7, 6, 3, 1}),
(21, {1, 3, 6, 10, 6, 3, 1}),
(25, {1, 3, 6, 10, 10, 6, 3, 1}),
(29, {1, 3, 6, 10, 12, 10, 6, 3, 1}),
(32, {1, 3, 6, 10, 12, 12, 10, 6, 3, 1}),
(33, {1, 3, 6, 10, 15, 10, 6, 3, 1}),
(38, {1, 3, 6, 10, 15, 15, 10, 6, 3, 1}),
(45, {1, 3, 6, 10, 15, 19, 15, 10, 6, 3, 1})};
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