CSM.m2 is a collection of Macaulay2 functions computing (the push-forward to projective space of) certain classes determined by a projective scheme S. The input to all functions is the defining ideal of S in projective space: it is expected that this is a homogeneous ideal in a polynomial ring. The output consists of polynomials in a variable H; these are to be interpreted as elements in the Chow ring of projective space. This ring is globally defined as intringPn by the routine. The functions print out their result to the screen, and store it in global variables.
The examples that follow are meant to be transparent both in the mathematics and in the use of the code. Suggestions for further examples are very welcome.
I take this opportunity to thank Dan Grayson and Mike Stillman for creating the excellent Macaulay2.
themis{aluffi}1: Macaulay2
Macaulay 2, version 0.9
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 1.3b, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 0.3.2, copyright 1996-2001, M. Messollen
|
We start by loading CSM, and defining a few rings for the examples that follow.
i1 : load "CSM.m2" --loaded CSM.m2 i2 : ringP2=QQ[x,y,z]; ringP3=QQ[x,y,z,w]; ringP4=QQ[x,y,z,w,t]; |
Define the ideal of S1=three distinct, concurrent, non-coplanar lines in P3, and compute its Segre class in P3:
i5 : use ringP3; threelines=ideal(x*y,x*z,y*z);
o6 : Ideal of ringP3
i7 : segre threelines
3 2
.Segre class : - 10H + 3H
|
The `.' signals that one random choice for a `general' hyperplane had to be rejected. Next we define the ideal of S2=three (distinct, concurrent) coplanar lines in P3, and compute its Segre class.
i8 : threecoplanarlines=ideal(z,x*y*(x+y));
o8 : Ideal of ringP3
i9 : segre threecoplanarlines
3 2
Segre class : - 12H + 3H
|
Both the Segre class and the Fulton class of the scheme are now available as global variables. For example, by the result of Example 4.2.6(a) in Fulton's Intersection theory, the following must equal the Segre class of S2 in P2:
i10 : (1+H)*segreclass
3 2
o10 = - 9H + 3H
o10 : intringPn
|
This is written as an element of the Chow ring of P3, with H as the hyperplane class; so it should be read as 3 times the class of a line, minus 9 times the class of a point. To perform directly the (trivial) computation of the Segre class in P2:
i11 : use ringP2; threecoplanarlines2=ideal(x*y*(x+y));
o12 : Ideal of ringP2
i13 : segre threecoplanarlines2
2
Segre class : - 9H + 3H
|
Note that the active intersection ring has changed, so now H denotes the hyperplane class in P2.
Not surprisingly, computations in P2 take less than in P3, even when they compute `the same' object (these times were obtained on a SunFire V880, running Solaris 8, in 2002):
i14 : time CF threecoplanarlines2
Fulton class : 3H
-- used 0.4 seconds
i15 : time CF threecoplanarlines
2
Fulton class : 3H
-- used 4.71 seconds
|
As the schemes are isomorphic, and the Fulton class is intrinsic, these computation agree. Again, the difference in the output is due to the fact that the first one is given in the Chow ring of P2, while the second one is given in the Chow ring of P3.
The Fulton class of the two (nonisomorphic) schemes S1 and S2 differ:
i16 : CF threelines
3 2
Fulton class : 2H + 3H
i17 : CF threecoplanarlines
2
Fulton class : 3H
|
By contrast, the Chern-Schwartz-MacPherson classes agree:
i18 : CSM threelines
3 2
Chern-Schwartz-MacPherson class : 4H + 3H
i19 : CSM threecoplanarlines
3 2
Chern-Schwartz-MacPherson class : 4H + 3H
|
The Euler characteristic is the coefficient of H3, that is (of course) 4.
For an example of two nonisomorphic schemes with the same Fulton class, but different Chern-Schwartz-MacPherson classes, consider the union of three planes in P3, vs. a nonsingular cubic surface:
i20 : use ringP3;threeplanes=ideal(x*y*z);cubic=ideal(x^3+y^3+z^3+w^3);
o21 : Ideal of ringP3
o22 : Ideal of ringP3
i23 : CF threeplanes; CF cubic;
3 2
Fulton class : 9H + 3H + 3H
3 2
Fulton class : 9H + 3H + 3H
i25 : CSM threeplanes; CSM cubic;
3 2
Chern-Schwartz-MacPherson class : 4H + 6H + 3H
3 2
Chern-Schwartz-MacPherson class : 9H + 3H + 3H
|
The Fulton and Chern-Schwartz-MacPherson classes agree for the second scheme since it is nonsingular. The Euler characteristic is computed to 9, as it should. To compute the Euler characteristic of the (complex) affine nonsingular cubic surface x3+y3+z3=1:
i27 : use QQ[x,y,z]; euleraffine ideal(x^3+y^3+z^3-1) o28 = 9 |
This is also 9, as the cubic curve `at infinity' is nonsingular. Not so for the affine nonsingular cubic surface x3+y3+z2=1, and the Euler characteristic drops accordingly:
i29 : euleraffine ideal(x^3+y^3+z^2-1) o29 = 5 |
For an example with nilpotents, here are the Fulton and Chern-Schwartz-MacPherson classes for the scheme with ideal (x,z)(y,z) in P3, supported on two lines:
i30 : use ringP3; twolinesnilp=ideal(x,z)*ideal(y,z);
o31 : Ideal of ringP3
i32 : CF twolinesnilp; CSM twolinesnilp;
3 2
Fulton class : 4H + 2H
3 2
Chern-Schwartz-MacPherson class : 3H + 2H
|
The Fulton class `sees' the nilpotent, while the Chern-Schwartz-MacPherson does not; compare with the computation for the reduced scheme supported on the same two lines:
i34 : twolinesred=ideal(x*y,z);
o34 : Ideal of ringP3
i35 : CF twolinesred; CSM twolinesred;
3 2
Fulton class : 2H + 2H
3 2
Chern-Schwartz-MacPherson class : 3H + 2H
|
The Euler characteristic of a nonsingular quintic threefold is -200:
i37 : use ringP4; CSM ideal(x^5+y^5+z^5+w^5+t^5)
4 3
Chern-Schwartz-MacPherson class : - 200H + 50H + 5H
|
And here is the computation for the (singular) quintic obtained by closing in P4 the elliptic fibration y2=x3+xz4+w5 in A4:
i39 : time CSM ideal(x^3*t^2+x*z^4+w^5-y^2*t^3)
4 3
Chern-Schwartz-MacPherson class : 4H + 38H + 5H
-- used 15.19 seconds
|
Examples in positive characteristic: they refer to the ideal dtupleideal, whose definition is available here. This is the ideal of the base scheme of a certain rational map encountered in studying the orbit closure of the 5-tuple of points on P1 defined by (s(s+3t)2(s+5t)(s+16t)) (see the paper, section 4, for further details). The Segre classes in characteristic 2 and 3 resp. are obtained by:
i40 : use ZZ/2[x,y,z,w]; load "dtupleideal.m2"; time segre dtupleideal
--loaded dtupleideal.m2
3 2
..Segre class : - 70H + 13H
-- used 2.48 seconds
i43 : use ZZ/3[x,y,z,w]; load "dtupleideal.m2"; time segre dtupleideal
--loaded dtupleideal.m2
3 2
.Segre class : - 58H + 11H
-- used 3.02 seconds
|
The computation in higher characteristic takes a substantially longer time (about 20 minutes for characteristic 5). These classes encode enumerative information; for example, since
i46 : (1+5*H)^3*segreclass
3 2
o46 = 107H + 11H
o46 : intringPn
|
it follows that the predegree of the orbit closure of the 5-tuple given above equals 53-107=18 in characteristic 3.
The Milnor class measures the difference between the Fulton and the Chern-Schwartz-MacPherson class. This class carries good information about the singularities. For a hypersurface with isolated singularities, this amounts to the Milnor number (hence the name).
i47 : use ringP2; milnor ideal(y^6+z*x^3*y^2+z^2*x^4)
2
Fulton class : - 18H + 6H
Chern-Schwartz-MacPherson class : 6H
2
Milnor class : 18H
|
This says that the sum of the Milnor numbers of the singularities of the curve y6+x3y2z+ x4z2=0 in P2 is 18. For arbitrary singularities of hypersurfaces in Pn, the coefficient of Hn in the Milnor class equals (up to sign) the sum of Parusinski's generalization of the Milnor number. For example, the cone y2z-x3-x2z in P3 is singular along a line:
i49 : use ringP3; top ideal jacobian ideal(y^2*z-x^3-x^2*z) o50 = ideal (y, x) o50 : Ideal of ringP3 |
The Milnor class in this example is the class of the line, minus 7 points:
i51 : milnor ideal(y^2*z-x^3-x^2*z)
3 2
Fulton class : 9H + 3H + 3H
3 2
Chern-Schwartz-MacPherson class : 2H + 4H + 3H
3 2
Milnor class : - 7H + H
|
and it follows that the Parusinski-Milnor number of this singularity is 7.