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distinguished -- compute the distinguished subvarieties of a scheme

Synopsis

Description

Let I⊂R be an ideal in a ring R, the image of a free R-module F. Let ReesI be the Rees algebra of I. Certain of the minimal primes of I are distinguished from the point of view of intersection theory: These are the ones that correspond to primes P⊂ReesI minimal among those containing I*ReesI---in other words, the isolated components of the support of the normal cone of I. The prime p corresponding to P is simply the kernel of the the induced map R →ReesI/P.

Each of these primes comes equipped with a multiplicity, which may be computed as the ratio degree(ReesI/P)/degree(R/p).

For these matters and their significance, see section 6.1 of the book “Intersection Theory,” by William Fulton, and the references there, along with the paper

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

i1 : T = ZZ/101[c,d];
i2 : D = 4;
i3 : P = product(D, i -> random(1,T))

          4     3       2 2       3      4
o3 = - 50c  - 2c d + 31c d  - 7c*d  - 36d

o3 : T
i4 : R = ZZ/101[a,b,c,d]

o4 = R

o4 : PolynomialRing
i5 : I = ideal(a^2, a*b*(substitute(P,R)), b^2)

             2           4         3           2 2           3          4 
o5 = ideal (a , - 50a*b*c  - 2a*b*c d + 31a*b*c d  - 7a*b*c*d  - 36a*b*d ,
     ------------------------------------------------------------------------
      2
     b )

o5 : Ideal of R
There is one minimal associated prime (a thick line in P3) and D embedded primes (points on the line).
i6 : ass I

o6 = {ideal (b, a), ideal (c + 39d, b, a), ideal (c + 32d, b, a), ideal (c -
     ------------------------------------------------------------------------
     13d, b, a)}

o6 : List
i7 : primaryDecomposition I

              2        2                    2   2                    2   2  
o7 = {ideal (b , a*b, a ), ideal (c + 32d, b , a ), ideal (c - 13d, b , a ),
     ------------------------------------------------------------------------
             2             2   2   2
     ideal (c  - 23c*d + 6d , b , a )}

o7 : List
Only the minimal prime is a distinguished component, and it has multiplicity 2.
i8 : distinguished(I)

o8 = {ideal (b, a)}

o8 : List
i9 : K = distinguishedAndMult(I)

o9 = {{2, ideal (b, a)}}

o9 : List

See also

  • distinguishedAndMult -- compute the distinguished subvarieties of a scheme along with their multiplicities

Ways to use distinguished :

  • distinguished(Ideal)
  • distinguished(Ideal,RingElement)