This is an example of application of the method projectiveDegrees; see Proposition 4.4 in Intersection theory, by W. Fulton, and Subsection 2.3 in Lectures on Cremona transformations, by I. Dolgachev. See also the corresponding methods in the packages CSM-A, by P. Aluffi, and CharacteristicClasses, by M. Helmer and C. Jost.
In the example below, we take $Y\subset\mathbb{P}^7$ to be the dual hypersurface of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset{\mathbb{P}^7}^*$ and $X\subset Y$ its singular locus. We compute the push-forward to the Chow ring of $\mathbb{P}^7$ of the Segre class both of $X$ in $Y$ and of $X$ in $\mathbb{P}^7$, using both a probabilistic and a non-probabilistic approach.
i1 : P7 = ZZ/100003[x_0..x_7] o1 = P7 o1 : PolynomialRing |
i2 : Y = ideal(x_3^2*x_4^2-2*x_2*x_3*x_4*x_5+x_2^2*x_5^2-2*x_1*x_3*x_4*x_6-2*x_1*x_2*x_5*x_6+4*x_0*x_3*x_5*x_6+x_1^2*x_6^2+4*x_1*x_2*x_4*x_7-2*x_0*x_3*x_4*x_7-2*x_0*x_2*x_5*x_7-2*x_0*x_1*x_6*x_7+x_0^2*x_7^2)
2 2 2 2 2 2
o2 = ideal(x x - 2x x x x + x x - 2x x x x - 2x x x x + 4x x x x + x x
3 4 2 3 4 5 2 5 1 3 4 6 1 2 5 6 0 3 5 6 1 6
------------------------------------------------------------------------
2 2
+ 4x x x x - 2x x x x - 2x x x x - 2x x x x + x x )
1 2 4 7 0 3 4 7 0 2 5 7 0 1 6 7 0 7
o2 : Ideal of P7
|
i3 : X = sub(ideal jacobian Y,P7/Y)
2
o3 = ideal (4x x x - 2x x x - 2x x x - 2x x x + 2x x , - 2x x x -
3 5 6 3 4 7 2 5 7 1 6 7 0 7 3 4 6
------------------------------------------------------------------------
2 2
2x x x + 2x x + 4x x x - 2x x x , - 2x x x + 2x x - 2x x x +
2 5 6 1 6 2 4 7 0 6 7 3 4 5 2 5 1 5 6
------------------------------------------------------------------------
2 2
4x x x - 2x x x , 2x x - 2x x x - 2x x x + 4x x x - 2x x x , 2x x
1 4 7 0 5 7 3 4 2 4 5 1 4 6 0 5 6 0 4 7 3 4
------------------------------------------------------------------------
2
- 2x x x - 2x x x + 4x x x - 2x x x , - 2x x x + 2x x - 2x x x +
2 3 5 1 3 6 1 2 7 0 3 7 2 3 4 2 5 1 2 6
------------------------------------------------------------------------
2
4x x x - 2x x x , - 2x x x - 2x x x + 4x x x + 2x x - 2x x x ,
0 3 6 0 2 7 1 3 4 1 2 5 0 3 5 1 6 0 1 7
------------------------------------------------------------------------
2
4x x x - 2x x x - 2x x x - 2x x x + 2x x )
1 2 4 0 3 4 0 2 5 0 1 6 0 7
P7
o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2
x x - 2x x x x + x x - 2x x x x - 2x x x x + 4x x x x + x x + 4x x x x - 2x x x x - 2x x x x - 2x x x x + x x
3 4 2 3 4 5 2 5 1 3 4 6 1 2 5 6 0 3 5 6 1 6 1 2 4 7 0 3 4 7 0 2 5 7 0 1 6 7 0 7
|
i4 : time SegreClass X
-- used 0.537771 seconds
7 6 5 4 3
o4 = 3240H - 1188H + 396H - 114H + 24H
ZZ[H]
o4 : -----
8
H
|
i5 : time SegreClass lift(X,P7)
-- used 0.432733 seconds
7 6 5 4 3
o5 = 2816H - 1056H + 324H - 78H + 12H
ZZ[H]
o5 : -----
8
H
|
i6 : time SegreClass(X,MathMode=>true)
MathMode: output certified!
-- used 0.0280497 seconds
7 6 5 4 3
o6 = 3240H - 1188H + 396H - 114H + 24H
ZZ[H]
o6 : -----
8
H
|
i7 : time SegreClass(lift(X,P7),MathMode=>true)
MathMode: output certified!
-- used 0.138098 seconds
7 6 5 4 3
o7 = 2816H - 1056H + 324H - 78H + 12H
ZZ[H]
o7 : -----
8
H
|
i8 : o4 == o6 and o5 == o7 o8 = true |
The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
i9 : use ZZ/100003[x_0..x_6]
ZZ
o9 = ------[x ..x ]
100003 0 6
o9 : PolynomialRing
|
i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
-- used 0.0774273 seconds
ZZ
------[y ..y ]
100003 0 9 ZZ 2 2
o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y - y y - y y , y y - y y , y - y y - y y , y y + y y - y y , y y - y y , y y - y y - y y , y y - y y - y y })
(y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y ) 100003 0 6 3 0 5 1 6 3 4 1 7 4 2 7 0 9 2 5 3 5 1 8 4 5 1 9 4 8 2 9 3 9 7 8 4 9 6 9
5 7 4 8 2 9 5 6 3 8 1 9 4 6 3 7 0 9 2 6 1 7 0 8 2 3 1 4 0 5
ZZ
------[y ..y ]
100003 0 9 ZZ
o10 : RingMap ---------------------------------------------------------------------------------------------------- <--- ------[x ..x ]
(y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y ) 100003 0 6
5 7 4 8 2 9 5 6 3 8 1 9 4 6 3 7 0 9 2 6 1 7 0 8 2 3 1 4 0 5
|
i11 : time SegreClass phi
-- used 0.240119 seconds
9 8 7 6 5
o11 = 23H - 42H + 36H - 22H + 9H
ZZ[H]
o11 : -----
10
H
|
i12 : B = ideal matrix phi
2 2
o12 = ideal (y - y y - y y , y y - y y , y - y y - y y , y y + y y -
3 0 5 1 6 3 4 1 7 4 2 7 0 9 2 5 3 5
-----------------------------------------------------------------------
y y , y y - y y , y y - y y - y y , y y - y y - y y )
1 8 4 5 1 9 4 8 2 9 3 9 7 8 4 9 6 9
ZZ
------[y ..y ]
100003 0 9
o12 : Ideal of ----------------------------------------------------------------------------------------------------
(y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y , y y - y y + y y )
5 7 4 8 2 9 5 6 3 8 1 9 4 6 3 7 0 9 2 6 1 7 0 8 2 3 1 4 0 5
|
i13 : -- Segre class of B in G(1,4)
time SegreClass B
-- used 0.279299 seconds
9 8 7 6 5
o13 = 23H - 42H + 36H - 22H + 9H
ZZ[H]
o13 : -----
10
H
|
i14 : -- Segre class of B in P^9
time SegreClass lift(B,ambient ring B)
-- used 0.9863 seconds
9 8 7 6 5
o14 = 2764H - 984H + 294H - 67H + 9H
ZZ[H]
o14 : -----
10
H
|
The object SegreClass is a method function with options.