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GlicciPointsInP3 :: checkCandidates

checkCandidates -- check which numerical candidates lead to a bi-dominant Gorenstein linkage correspondence

Synopsis

Description

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

The failing candidates of the list of all candidates are
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})};

The cases in which the projection HCX ⊂G →HCX is 1-1 are the following:
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})};

Ways to use checkCandidates :

  • checkCandidates(List,Ring)