distinguished -- Compute the distinguished subvarieties of a pullback, intersection or cone

Synopsis

• Usage:
L = distinguished(f,I)
L = distinguished(I,J)
L = distinguished(I)
• Inputs:
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• Variable => ..., default value w, Choose name for variables in the created ring
• Outputs:

Description

Suppose that f:S\to R is a map of rings, and I is an ideal of S. Let K be the kernel of the map of associated graded rings gr_I(S) \to gr_(fI)R.

The distinguished primes p_i in S/I are the intersections of the minimal primes P_i over K with S/I \subset{} gr_IS, that is, the minimal primes of the support in R/I of the normal cone of f(I). The multiplicity associated with p_i is by definition the the multiplicities of P_i in the primary decomposition of K.

Distinguished subvarieties and their multiplicity (defined by the distinguished primes, usually in the global case of a quasi-projective variety and its sheaf of rings) play a central role in the Fulton-MacPherson construction of refined intersection products. See William Fulton, Intersection Theory, Section 6.1 for the geometric context and the general case, and the explanation in the article Rees Algebras in JSAG (submitted).

This application is illustrated in the code for intersectInP.

We allow the special cases

distinguished(I,J) := distinguished(f,I), with f:S\to S/J the projection

and

distinguished(I) := distinguished(f,I), with f:S\to S the identity.

which computes the distinguished primes in the support of the normal cone gr_IS itself. An interesting application is given in the paper

A geometric effective Nullstellensatz,'' Invent. Math. 137 (1999), no. 2, 427–448 by Ein and Lazarsfeld.

Here is an example showing that associated primes need not be distinguished primes:

 i1 : R = ZZ/101[a,b] o1 = R o1 : PolynomialRing i2 : I = ideal(a^2, a*b) 2 o2 = ideal (a , a*b) o2 : Ideal of R i3 : ass I o3 = {ideal a, ideal (b, a)} o3 : List

There is one minimal associated prime (a thick line in $P^3$) and one embedded prime.

 i4 : distinguished I o4 = {{2, ideal (b, a)}, {1, ideal a}} o4 : List i5 : intersectInP(I,I) o5 = {{1, ideal a}} o5 : List