Given a saturated ideal J of a zero-dimensional subscheme, irrelevant ideal irr, and a tuple A, resolveViaFatPoint computes a free resolution of J intersected with A-th power of the irrelevant ideal. See Theorem 4.1 of [BES20, arXiv:1703.07631].
Below we follow example 4.7 of [BES20,arXiv:1703.07631] and compute the virtual resolution of 6 points in $\PP^1\times\PP^1\times\PP^2$.
i1 : N = {1,1,2}
o1 = {1, 1, 2}
o1 : List
|
i2 : pts = 6 o2 = 6 |
i3 : (S, E) = productOfProjectiveSpaces N o3 = (S, E) o3 : Sequence |
i4 : irr = intersect for n to #N-1 list (
ideal select(gens S, i -> (degree i)#n == 1)
);
o4 : Ideal of S
|
i5 : I = saturate intersect for i to pts - 1 list (
P := sum for n to N#0 - 1 list ideal random({1,0,0}, S);
Q := sum for n to N#1 - 1 list ideal random({0,1,0}, S);
R := sum for n to N#2 - 1 list ideal random({0,0,1}, S);
P + Q + R
);
o5 : Ideal of S
|
i6 : C = resolveViaFatPoint (I, irr, {2,1,0})
1 17 34 24 6
o6 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o6 : ChainComplex
|
i7 : isVirtual(irr, C) o7 = true |
The object resolveViaFatPoint is a method function.