The beta invariant of a flag and the conormal intersection theory In this section, we complete the proof that the degree of π ˚

px_Fqδn´k´1 is equal to the β-invariant βMrFs. To prove by induction, we need a lemma relating the conormal fan of M with that of the contraction M {i, where i is any fixed element of E that has no parallel elements in M.

We may assume that M has no loops and no coloops. Thus, we have iK_{is the ground set E} and i|E is a biflat of M. We consider the simplicial fan

st_i|EΣM,MK Ď pNE{ eiq ‘ NE.

We write eSfor the image of eSin NE{ ei, and xF |Gfor the variable in the Chow ring of the star corresponding to a biflat F |G; we set it equal to 0 if F |G does not correspond to a ray in this star. Lemma 4.17. Consider the natural projection ψ : pNE{ eiq ‘ NE ÝÑ NE´i‘ NE´i.

(1) The projection ψ induces a morphism of fans

ψ : st_i|EΣM,MK ÝÑ Σ_{M {i,pM {iq}K.

(2) The pullback ψ˚_{between the Chow rings is given by} ψ˚_px

P |Qq “ xpP Yiq|Q` xpP Yiq|pQYiq, where at least one of the terms in the right-hand side is nonzero. (3) For any element j of E, the pullback ψ˚_{maps the class δ to the class}

δ “ δj – X iPF, jPF XG

x_{F |G}.

(4) The pullback ψ˚_{commutes with the degree maps of the star and the conormal fan Σ}

M {i,pM {iqK:

deg_{M {i}x_P|Q“ degMpxi|Eψ˚pxP|Qqq.

Proof. Let F |G be a biflat of M with i P F . The image of a ray corresponding to F |G in the star is ψpeF` fGq “ eF ´i` fG´i,

which is a ray of the conormal fan ΣM {i,pM {iqK because pF ´ iq|pG ´ iq is a biflat of M {i:

cl_{M {i}pF ´ iq “ clMpF q ´ i “ F ´ i, and

cl_MK_´ipG ´ iq “ cl_MKpG ´ iq ´ i Ď cl_MKpGq ´ i “ G ´ i.

Furthermore, if i|E YF|G is a biflag of M, its gaps occur to the right of i|E, and there will also be gaps in the corresponding positions of the biflag

pF ´ iq|pG ´ iq –npF ´ iq|pG ´ iq o

F |GPF|G of M {i. Therefore, the projection ψ maps cones to cones. This proves the first statement.

The value of the piecewise linear function ψ˚_x

P |Qon a ray eF` fGof the star is

ψ˚_x

P |QpeF` fGq “ xP |QpeF ´i` fG´iqq “  



1 if F “ P Y i and G P {Q, Q Y i}, 0 if otherwise.

Since Q is a flat of M {i, at least one of Q and Q Y i is a flat of M, and we have the second statement. Given the second statement, the third statement is a straightforward computation

ψ˚ pδjq “ X jPP XQ  x_{pP Yiq|Q}` xpP Yiq|pQYiq  “ X iPF, jPF XG x_{F |G}“ δj,

where the first sum is over the biflats P |Q of M {i and the second sum over the biflats F |G of M. For the last statement, we need to verify that, for any maximal biflagP|Q of M {i,

deg_{M {i}x_P|Q“ degMpxi|Eψ˚pxP|Qqq.

Applying the second statement to each P |Q inP|Q, we may express xi|Eψ˚_px

P|Qqas a sum of square-free monomials. One of the terms in this expression is xi|Ex_{pPYiq| cl}K_pQq, where

pP Y iq| clKpQq – n

pP Y iq| clKpQq o

P |QPP|Q. We need to prove that this is the only nonzero term.

Consider any term xi|Ex_F|Gthat arises in the expression for xi|Eψ˚_px

P|Qq. We automatically have Fj “ PjY ifor all j, so it remains to prove that Gjis the closure of Qjin MK_{for all j. Let} kbe the largest index satisfying i P clK

pQkq, so that

clKpQjq “ QjY i for j ď k and clKpQjq “ Qj for j ą k. For j ď k, the set Qjis not a flat in MK_{, and hence Gj}

“ QjY i “ clKpQjq. Since Qkand Qk`1are flats of consecutive ranks in pM {iqK_{“ M}K

´i, the flats QkY i “ clKpQkqand Qk`1“ clKpQk`1q of MK_{also have consecutive ranks. Since Qk`1}

Y iis strictly between the two, it cannot be a flat of MK. Thus, we must have Gk`1 “ Qk`1, and hence Gj “ Qj “ clKpQjqfor j ą k. We conclude thatF|G “ pP Y iq| clK

pQq as desired. _

We can now give an intersection-theoretic interpretation of the beta invariant of a flag. To- gether with the Vanishing Lemma4.15, it gives the identity degpπ˚_px

Fq δn´k´1q “ β_MrFs. Proposition 4.18. For any strictly increasing flagF of k nonempty proper flats of M, we have

degpx_F|FKδn´k´1q “ β_MrFs.

Proof. We proceed by induction on k. The base case k “ 0 follows from Proposition4.9: degpδn´1q “ deg

 _X

BPβ-nbcpMq

x_β-conepBq“ βM.

When k is positive, write F |FK_{for the first biflat in}F|FK_{, and write}F|FK_{as the disjoint union} F |FK _\G|GK_{. We consider the contraction M {F , its flag of flatsG ´ F – {G ´ F }GPG, and the} corresponding biflag pG ´ F q|pG ´ F qK – n pG ´ F q|pGK´ F q o GPG.

The displayed description is justified by GK

´ F “ clpM {F qKppE ´ F q ´ pG ´ F qq. Clearly,

β_MrFs“ βM |F ¨ βpM {F qrG´F s.

We separately consider two cases, depending on the shortness of the first interval r∅, F s: Case 1. The flat F contains exactly one element i P E.

Recall that the beta invariant of r∅, is is equal to 1. Therefore, βMrFsis equal to βpM {iqrG´is “ degM {i xG´i|pG´iqKδpn´1q´pk´1q´1

M {i

by the inductive hypothesis, “ deg_M x_i|Eψ˚_px G´i|pG´iqKq δ_Mn´k´1q
by Lemma4.17.3and4.17.4, “ deg_M x_i|E Y GPG pxG|pGK_´iq` x_G|GKq δn´k´1_M
by Lemma4.17.2. By the Vanishing Lemma4.15, the right-hand side simplifies to

deg_M x_i|Ex_G|GKδ_Mn´k´1
“ deg_M x_F|FKδ_Mn´k´1
.

Case 2. The flat F contains more than one element.

We may assume the interval r∅, F s is coloopless by Lemma4.15. This means that the flat F is cyclic, that is, FK_{“ E ´ F. We then have the natural bijections}

φ1: n biflats of M |FoÝÑ n biflats F1 |G1of M with F1 Ď F and G1 Ě E ´ F o and φ2:nbiflats of M {FoÝÑ n biflats F1 |G1of M with F1 Ě F and G1 Ď E ´ F o ,

where φ1pA|Bq “ A|pB Y pE ´ F qqand φ2pA|Bq “ pA Y F q|B. The bijection φ1extends to the bi- jection between the biflags of M |F and the biflags of M that are supported on the corresponding set of biflats, and have a gap to the left of F |pE ´ F q. Similarly, the bijection φ2extends to the bi- jection between the biflags of M {F and the biflags of M that are supported on the corresponding set of biflats, and have a gap to the right of F |pE ´ F q.

We now compute the degree of xF|FKδn´k´1 using the following variant of the canonical

expansion of Definition4.1, which proceeds in two stages:

Stage 1. At each step, choose e to be the largest gap element that is in F , if there is one. Stage 2. At each step, choose e to be the largest gap element in E ´ F .

The first |F | ´ 2 steps of this computation will give xF|FK times the image under φ1 of the

canonical expansion of δ_{M |F}|F |´2. By Proposition4.9, there will be βM |F square-free monomials. Each such monomial will have a unique nonempty gap before F ; say it is Dj, between biflats Fj|Gjand Fj`1|Gj`1of M. The flats Fjand Fj`1have consecutive ranks, and the coflats Gjand G<sub>j`1</sub>have consecutive coranks. In step |F | ´ 1 of the computation, this gap Dj will be filled in a unique way by the biflat Fj`1|Gj. There will no longer be gap elements in F .

In step |F |, the computation will enter Stage 2 for each of the resulting βM |F monomials. The following p|E ´ F | ´ 1q ´ pk ´ 1q ´ 1 steps will compute the image under φ2of the canonical

expansion of xp_{G´F q|pG´F q}Kδ|F |´2

M {F . This expansion has βpM {F qrG´F s square-free monomials, by the inductive hypothesis.

This concludes the computation of x_F|FKδn´k´1. The result will be the sum of β_MrFssquare-

free monomials, as we wished to prove. 

Proposition 4.19. LetF “ {F1Ĺ ¨ ¨ ¨ Ĺ Fk} be a strictly increasing flag of flats of M. We have degpπ˚px_Fq δn´k´1q “ βMrFs.

Proof. Since π˚_px

Fq “P_{F|G biflag}x_F|G, this follows from Lemma4.15and Proposition4.18. 

4.5. A conormal interpretation of the Chern–Schwartz–MacPherson cycles. Recall that the k- dimensional Chern–Schwartz–MacPherson cycle of M is the Minkowski weight csmkpMqon the Bergman fan of M defined by the formula

csmkpMqpσ_Fq “ p´1qr´kβMrFs,

where σF is the k-dimensional cone corresponding to a flag of flatsF of M. Theorem1.1. When M has no loops and no coloops, we have

csmkpMq “ p´1qr´kπ˚pδn´k´1X 1M,MKq for 0 ď k ď r.

Proof. By Proposition4.19, Definition3.4, and the projection formula, we have

β_MrFs“ degpπ˚px_Fqδn´k´1q “ π˚px_Fqδn´k´1X 1M,MK “ π_˚pδn´k´1X 1_M,MKqpσFq.

The result then follows from the definition of the Chern–Schwartz-MacPherson cycle of M.  Theorem1.2. When M has no loops and no coloops, we have

χ_Mpq ` 1q “ r X

k“0

p´1qr´kdegpγkδn´k´1q qk.

Proof. We use [LdMRS20, Theorem 1.4], which states that χ_Mpq ` 1q “

r X

k“0

αkX csmkpMq qk.

The authors of [AB20] give a non-recursive proof of the identity using tropical intersection the- ory. For representable matroids, the identity was given earlier in [Alu13, Theorem 1.2]. By Theorem1.1and the projection formula, the k-th coefficient of the displayed polynomial is

αkX π˚pδn´k´1X 1M,MKq “ π_˚ π˚αkX pδn´k´1X 1_M,MKq

“ π˚ γ

k_δn´k´1

X 1M,MK
.