Segre classes

In this subsection we state a result of Aluffi [1] (Proposition2.2.1) which can be used to calculate Segre classes of projective varieties. When combined with result of Theorem2.3.1 this yields our algorithm to compute Segre classes of projective varieties described in Algorithm2.3.2. We also review several previous results on the computation of Segre classes, the first due to Aluffi[1] and the second due to Eklund, Jost and Peterson [8].

In (2.16) we make explicit the relationship between the projective degrees of a rational map and the degrees of the residual set considered in [8].

Aluffi[2] gives the following result which allows for the computation of the Segre class ofV inPnforVa subscheme ofPn.

Proposition 2.2.1 (Proposition 3.1 [2]). Let I = (f0, . . . , fm) ⊂ k[x0, . . . ,xn] be a homogeneous ideal defining a scheme V ⊂ _Pn and let h = c1(OPn(1)) be the class

of a hyperplane in A∗(Pn). Since I is homogeneous we may assume, without loss

of generality, that the deg(fi) = d for all i. Let φ : Pn d Pm be the rational map

specified by

p7→(f0(p) :· · · : fm(p)),

let(g0, . . . ,gn)be the projective degrees ofφand letΓφ ⊂ Pn×Pm be the graph of φ. Write[G]for the class of the shadow of the graph of the mapφ(see(2.12)), i.e.

[G]= g0+g1h+· · ·+gn−1hn−1+gnhn

in A∗(Pn)Z[h]/(hn+1). Then we have: s(V,Pn)=1−c(O(dh))−1·        n X i=0 gihi c(O(dh))i₎        =1− n X i=0 gihi (1+dh)i+1 ∈A ∗ (Pn) Z[h]/(hn+1).

To use Proposition2.2.1, Aluffi[2] notes thatΓφcan be obtained explicitly asΓφis

isomorphic to the blow-up of Pn alongV, and once Γφis known the class [G] can

be computed directly. Specifically the algorithm of Aluffiis as follows,

• obtain Γφ explicitly (by computing BlVPn Γφ, that is the blow-up of Pn alongV)

• intersectΓφwith general hyperplanes

• project the intersections down to Pn, and compute the degree of the projec- tions to obtain the class of the shadow of the graph, [G].

Hence the main computational cost for finding Segre classes using the method of [2] is that of finding the blow-up ofPnalongV.

Another method for computing Segre classes was given by Eklund, Jost and Peter- son [8]. This method does not use the relation between the class of the shadow of the graph [G] (see (2.12)) and the Segre class s(V,Pn); we summarize the result in Proposition2.2.2below.

Proposition 2.2.2(Theorem 3.2 [8]). Let V ⊂ _Pn _{be a subscheme of dimension}%

defined by a non-zero homogeneous ideal I = (f0, . . . , fm) ⊂ k[x0, . . . ,xn]with the generators fi having degree d. Let

s(V,Pn)= sn+· · ·+s0hn ∈A∗(Pn)Z[h]/(hn+1)

J =(γ1, . . . , γj)and let Rj ⊂Pnbe the subscheme defined by J : I∞. Then we have dj = deg(Rj)+ j−(n−%) X i=0 j j−(n−%)−i ! dj−(n−%)−isi.

To apply Proposition 2.2.2to compute s(V,Pn), Eklund, Jost and Peterson [8] use the following method.

• V = V(I), saydis the degree of the homogeneous generators ofI.

• Pick general degreedpolynomialsω1, . . . , ωj inI.

• For j=n−dimV =codim(V) to j=ndo:

◦ SetJ =(ω1, . . . , ωj) and letRjbe the scheme defined by J :I∞.

◦ Compute deg(Rj). ◦ Setp= j−codim(V), sp= dj−deg(Rj)− p−1 X i=1 j p−i ! dp−isi. (2.13)

Hence the main computational cost in the algorithm of Eklund, Jost and Peterson [8] is the computation of deg(Rj). When done symbolically, this means the main cost arises from the computation of the saturationJ :I∞for each j. Eklund, Jost and Peterson [8] also explain that deg(Rj) can be computed numerically using homotopy continuation in Bertini [5].

There is, in fact, an explicit relationship between the projective degrees (g0, . . . ,gn) of a rational mapφdefined by an idealI(or equivalently the class [G] of the shadow of the graphΓφ(2.12)) and the degrees of the residual setsRj in Proposition2.2.2. Specifically letV =V(I) be a subscheme of_Pn_where

I =(f0, . . . , fm) is a homoge- neous ideal ink[x0, . . . ,xn] and let [G]= g0+g1h+· · ·+gn−1hn−1+gnhn ∈A∗(Pn) be

the class of the shadow of the graph ofφ(as in Proposition2.2.1). SinceIis homo- geneous we may assume that deg(fj) = dfor alli = 1, . . . ,m. Takeν = codim(Y). Let

s(V,Pn)= sn+· · ·+s0hn ∈A∗(Pn)

be the Segre class ofV inPn and let ˜s0 = 1, ˜s1 = · · · = s˜ν−1 = 0 and ˜si = −si−ν for

i ≥ ν. Note that sn = · · · = sν+1 = 0, i.e. sν is the first nonzero coefficent. In [17] Jost gives the following expression relating thegjin the class of the graphΓI to the Segre class, gj = j X i=0 j i ! dj−is˜i, (2.14) which is obtained by rearranging and simplifying the expression of Proposition 2.2.1. The result of Proposition 2.2.2 gives the following expression for deg(Rj) when j=ν, . . . ,n, deg(Rj)= dj− j−(n−ν) X i=0 j j−(n−ν)−i ! dj−(n−ν)−isi. (2.15)

Reindexing the summation in (2.15) we have

deg(Rj)=dj− j X i=ν j i ! dj−isi−ν, for j= ν, . . . ,n.

Since ˜s0 = 1 and ˜s1 = · · · = s˜ν−1 = 0 we may rewrite the expression (2.14) forgj as gj =dj− j X i=ν j i ! dj−isi−ν, for j= ν, . . . ,n,

andgj = djfor j= 0, . . . , ν−1. Hence we have that

deg(Rj)=gj for j=ν, . . . ,n. (2.16) In light of (2.16) we observe that the method for computing Segre classes of Eklund, Jost and Peterson [8] stated in Proposition 2.2.2computes the same values as the

result of Theorem2.3.1, and in fact, the method of Theorem2.3.1 can be seen as a refinement of the method of [8]. In both cases similar systems of equations are considered, however we will see below that the method of Algorithm2.3.1tends to perform better.