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sectionRing -- sectionRing(I) produces the section ring of an ample divisor. If I is an ideal, one can input I to get the section ring of the corresponding divisor.



sectionRing(I) begins by computing the regularity m of O_X, O_X(D), O_X(2D), ..., O_X((l-1)D) with respect to O_X(lD), where l is the output of globallyGenerated(D). By Mumford's Thm (1.8.5 in Positivity) yields that each of the maps O_X(iD)\otimes O_X(lD)^\otimes m -> O_X((i+ml)D) is surjective. Thus, all generators for the section ring can be assumed in lower degree than bound. Thus it forms a polynomial ring S over the base field with h^0(iD)-many generators in degree i, for i=1,2,...,bound-1. Next, relations in degree d are computing by considering the total maps \oplus_{partitions P of d} \otimes_{i\in P} O_X(i D) -> O_X(dD). Each of these relations is then quotiented, until the point that a domain of the correct dimension is produced. Some steps are then performed to make the output more readable and standard.

Ways to use sectionRing :

For the programmer

The object sectionRing is a method function.