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VirtualResolutions :: resolveViaFatPoint

resolveViaFatPoint -- returns a virtual resolution of a zero-dimensional scheme

Synopsis

Description

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 [BES, arXiv:1703.07631].

Below we follow example 4.7 of [BES,arXiv:1703.07631] and compute the virtual resolution of 6 points in 1×ℙ1×ℙ2.

N = {1,1,2}
pts = 6
(S, E) = productOfProjectiveSpaces N
irr = intersect for n to #N-1 list ( ideal select(gens S, i -> (degree i)#n == 1) );
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 );
C = resolveViaFatPoint (I, irr, {2,1,0})
isVirtual(I, irr, C)

Ways to use resolveViaFatPoint :