# SegreClass -- Segre class of a closed subscheme of a projective variety

## Synopsis

• Usage:
SegreClass I
• Inputs:
• I, an ideal, a homogeneous ideal of a graded quotient ring $K[x_0,\ldots,x_n]/J$ defining a subscheme $X=V(I)$ of $Y=Proj(K[x_0,\ldots,x_n]/J)\subseteq \mathbb{P}^n=Proj(K[x_0,\ldots,x_n])$
• Optional inputs:
• BlowUpStrategy => ..., default value "Eliminate",
• Certify => ..., default value false, whether to ensure correctness of output
• Verbose => ..., default value true,
• Outputs:
• , the push-forward to the Chow ring of $\mathbb{P}^n$ of the Segre class $s(X,Y)$ of $X$ in $Y$

## Description

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 1.57109 seconds 7 6 5 4 3 o4 = 3240H - 1188H + 396H - 114H + 24H ZZ[H] o4 : ----- 8 H i5 : time SegreClass lift(X,P7) -- used 1.27418 seconds 7 6 5 4 3 o5 = 2816H - 1056H + 324H - 78H + 12H ZZ[H] o5 : ----- 8 H i6 : time SegreClass(X,Certify=>true) Certify: output certified! -- used 0.0894369 seconds 7 6 5 4 3 o6 = 3240H - 1188H + 396H - 114H + 24H ZZ[H] o6 : ----- 8 H i7 : time SegreClass(lift(X,P7),Certify=>true) Certify: output certified! -- used 0.383137 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 0.276223 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 0.838104 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 0.823909 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 3.00426 seconds 9 8 7 6 5 o14 = 2764H - 984H + 294H - 67H + 9H ZZ[H] o14 : ----- 10 H