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

Synopsis

Usage:

resolveViaFatPoint(I, irr, A)
Inputs:
- J, an ideal, saturated ideal corresponding to a zero-dimensional scheme
- irr, an ideal, the irrelevant ideal
- A, a list, power you want to take the irrelevant ideal to
Outputs:
- a chain complex, virtual resolution of our ideal

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 [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

Ways to use resolveViaFatPoint :

resolveViaFatPoint(Ideal,Ideal,List)

For the programmer

The object resolveViaFatPoint is a method function.