Given a module M over a multigraded ring S or a product of toric varieties X, this method finds the minimal elements of the multigraded Castelnuovo-Mumford regularity of M as defined in Definition 1.1 of [MS04] (see arXiv:math/0305214). If the input is an ideal, multigraded regularity of $S^1/I$ is computed.
There are two strategies implemented and run using hooks:
Note that both strategies require the module or ideal to be saturated by the irrelevant ideal of the Cox ring.
As an example, here we compute the minimal elements of the multigraded regularity for Example 1.4 of [BES20] (see arXiv:1703.07631). We consider the example of a hyperelliptic curve of genus 4 in $\PP^1\times\PP^2$.
i1 : X = toricProjectiveSpace(1)**toricProjectiveSpace(2) o1 = X o1 : NormalToricVariety |
i2 : S = ring X; B = ideal X; o3 : Ideal of S |
i4 : I = ideal(x_0^2*x_2^2+x_1^2*x_3^2+x_0*x_1*x_4^2, x_0^3*x_4+x_1^3*(x_2+x_3)) 2 2 2 2 2 3 3 3 o4 = ideal (x x + x x + x x x , x x + x x + x x ) 0 2 1 3 0 1 4 1 2 1 3 0 4 o4 : Ideal of S |
After saturating the defining ideal by the irrelevant ideal we may compute its multigraded regularity.
i5 : J = saturate(I,B); o5 : Ideal of S |
i6 : L = multigradedRegularity(X, J) o6 = {{2, 2}, {4, 1}, {1, 5}} o6 : List |
If debugLevel is larger than zero, additional information about the degree search is printed.
This method also accepts the ring provided by productOfProjectiveSpaces from the TateOnProducts package.
Lauren Cranton Heller contributed to the code for this method.
The input is assumed to be saturated. Moreover, if the input is a module generated in non-positive degrees, then the output may be incorrect. In that case, adding the optional argument
LowerLimit => apply(n, i -> min(degrees M / (deg -> deg_i))) - dim X
where M is the module and X is the toric variety, may be a sufficient solution.
The object multigradedRegularity is a method function with options.