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iteratedCone -- Computes a (possibly non-minimal) resolution of C in P^{g-1} starting from the relative canonical resolution of C in P(E)

Synopsis

Description

Given the relative canonical resolution of C on a normalized scroll $P(E)$, this function computes a (possibly non-minimal) free resolution of C in $P^{g-1}$ by an iterated mapping cone construction. For gonality k=3,4 the iterated mapping cone is always minimal. In these cases "iteratedCone" is much faster (for $g >9$) than computing the resolution via the resolution command.

i1 : (g,k,n) = (8,5,1000)

o1 = (8, 5, 1000)

o1 : Sequence
 
i2 : e = balancedPartition(k-1,g-k+1)

o2 = {1, 1, 1, 1}

o2 : List
 
i3 : Ican = canCurveWithFixedScroll(g,k,n);

               ZZ
o3 : Ideal of ----[t ..t ]
              1009  0   7
 
i4 : betti res(Ican,DegreeLimit=>1)

            0  1  2  3
o4 = total: 1 15 35 21
         0: 1  .  .  .
         1: . 15 35 21

o4 : BettiTally
 
i5 : Jcan = curveOnScroll(Ican,g,k);

               ZZ
o5 : Ideal of ----[pp ..pp , v..w]
              1009   0    3
 
i6 : betti(resX = resCurveOnScroll(Jcan,g,2))

            0 1 2 3
o6 = total: 1 5 5 1
         0: 1 . . .
         1: . . . .
         2: . 4 1 .
         3: . 1 4 .
         4: . . . .
         5: . . . 1

o6 : BettiTally
 
i7 : betti(resC = iteratedCone(resX,e))

            0  1  2  3  4  5 6
o7 = total: 1 15 41 54 41 15 1
         0: 1  .  .  .  .  . .
         1: . 15 35 27  6  . .
         2: .  .  6 27 35 15 .
         3: .  .  .  .  .  . 1

o7 : BettiTally

See also

Ways to use iteratedCone :

For the programmer

The object iteratedCone is a method function.