Tropical Discriminants

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by  Alicia Dickenstein

JOURNALOFTHE
AMERICANMATHEMATICALSOCIETY
Volume00,Number0,Pages000–000
S0894-0347(XX)0000-0
TROPICAL DISCRIMINANTS
7
0
ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
0
2
Dedicated to the memory of Pilar Pis´on Casares

n
a
J

3 1. Introduction
1
Let A be an integer d×n-matrix such that (1,1,...,1) is in the row span of A.

] This defines a projective toric variety X in CPn−1. Its dual variety X∗ is the clo-
G A A
sureintheprojectivespacedualtoCPn−1ofthesetofhyperplanesthataretangent
A
to X at a regular point. The toric variety X is called non-defective if its dual
A A
h. varietyXA∗ hascodimensionone. Inthiscase,theA-discriminant istheirreducible
t homogeneous polynomial ∆ which defines the hypersurface X∗. Alternatively,
a A A
m the dualvariety XA∗ canbe thoughtofas the set ofsingularhypersurfacesin (C∗)d
withNewtonpolytopeprescribedbythematrixA. Thestudyoftheseobjectsisan
[
active area of researchin computational algebraic geometry, with the fundamental

3 reference being the monograph by Gel’fand, Kapranov and Zelevinsky [13].
v Our main object of interest in this paper is the tropical A-discriminant τ(X∗).
6 A
This is the polyhedralfan in Rn which is obtained by tropicalizing X∗. While it is
2 A
generally difficult to compute the dual variety X∗ from A, we show that its trop-
1 A
0 icalization τ(X∗) can be computed much more easily. In Theorem 1.1, we derive
A
1 an explicit description of the tropical A-discriminant τ(X∗), and in Theorem 1.2
A
5 we present an explicit combinatorial formula for the extreme monomials of ∆ .
A
0
Withoutlossofgenerality,weassumethatthecolumnsofthematrixAspanthe
/
h integer lattice Zd, and that the point configuration given by the columns of A is
t
a not a pyramid. These hypotheses ensure that the toric variety XA has dimension
m d−1 and that the dual variety X∗ is not contained in any coordinate hyperplane.
A
: A key player in this paper is the tropicalization of the kernel of A. As shown in
v
[1]and[10],thistropicallinearspaceissubdividedbothbytheBergman fan ofthe
i
X matroiddualto A, i.e., the co-Bergman fan B∗(A) ofA, andby the nested set fans
r of its lattice of flats L(A). In other words, let L(A) denote the geometric lattice
a whose elements are the sets of zero-entries of the vectors in kernel(A), ordered by
inclusion. WewriteC(A)forthesetofpropermaximalchainsinL(A)andrepresent
these chains as (n−d−1)-elementsubsets σ ={σ ,...,σ } of {0,1}n.
1 n−d−1
ReceivedbytheeditorsDecember18,2006.
2000 Mathematics Subject Classification. Primary14M25,Secondary52B20.
Key words and phrases. Tropicalgeometry, dualvariety,discriminant.
A.DickensteinwaspartiallysupportedbyUBACYTX042,CONICETPIP5617andANPCYT
17-20569,Argentina.
E.M.FeichtnerwassupportedbyaResearchProfessorshipoftheSwissNationalScienceFoun-
dation,PP002–106403/1.
B.SturmfelswaspartiallysupportedbytheU.S.NationalScienceFoundation,DMS-0456960.
(cid:13)c1997AmericanMathematicalSociety
1
2 ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
We obtain the following descriptions of the tropicalization of the kernel of A:
(1.1) τ(kernel(A)) = support(B∗(A)) = R σ.
≥0
σ∈[C(A)
Theunionontherighthandsideinfactisthefinestinthehierarchyofunimodular
simplicial fan structures provided by the nested set fans [5, 8, 9, 10, 11]. The
tropicallinear space (1.1) is a subsetof Rn. We obtainthe tropicalA-discriminant
byaddingthistropicallinearspacetothe(classical)rowspaceofthed×n-matrixA:
Theorem1.1. Foranyd×n-matrixAasabove, thetropicalA-discriminantτ(X∗)
A
equals the Minkowski sum of the co-Bergman fan B∗(A) and the row space of A.
Theorem 1.1 is the tropical analogue of Kapranov’s Horn uniformization [16].
By definition, the tropical discriminant τ(X∗) inherits the structure of a fan from
A
the Gr¨obner fan of the ideal of X∗ and, in the non-defective case, also from the
A
secondary fan of A [13]. In general, neither of these two fan structures refines the
other, as we shall see in Examples 5.5 and 5.6.
The tropicalization of a variety retains a lot of information about the geometry
of the original variety [18, 20, 21, 23, 24].
In Theorem 1.2 below, our tropical approachleads to a formula for the extreme
monomials of the A-discriminant ∆ , and, a fortiori, for the degree of the dual
A
variety X∗. An alternating product formula for the extreme monomials of ∆
A A
wasgivenin[13,§11.3.C]undertherestrictiveassumptionthatX issmooth. Our
A
formula(1.2)ispositive,itisvalidforanytoricvarietyX regardlessofsmoothness,
A
and its proof is self-contained.
Theorem 1.2. If X is non-defective and w a generic vector in Rn then the
A
exponent of x in the initial monomial in (∆ ) of the A-discriminant ∆ equals
i w A A
(1.2) | det(At,σ ,...,σ ,e )|,
1 n−d−1 i
σ∈XCi,w
whereC isthesubsetofC(A)consistingofallchainssuchthattherowspaceofthe
i,w
matrix A has non-empty intersection with the cone R σ ,...,σ ,−e ,−w .
>0 1 n−d−1 i
(cid:8) (cid:9)
Here, the A-discriminant ∆ is written as a homogeneous polynomial in the vari-
A
ables x1,...,xn, and inw(∆A) is the w-lowest monomial xu11···xunn which appears
intheexpansionof∆ incharacteristiczero. Theorem1.2generalizestothedefec-
A
tivecase,whenwetake∆ astheChow formofthedualvarietyX∗. Thisisstated
A A
inTheorem4.6. Aimingformaximalefficiencyinevaluating(1.2)withacomputer,
we can replace C with the corresponding maximal nested sets of the geometric
i,w
lattice L(A), or with the corresponding maximal cones in the Bergmanfan B∗(A).
Our maple implementation of the formula (1.2) is discussed in Section 5.
This paper is organized as follows. In Section 2, we review the construction of
the tropicalization τ(Y) of a projective variety Y, and we show how the algebraic
cycle underlying any initial monomial ideal of Y can be read off from τ(Y). In
Section 3, we discuss general varieties which are parametrized by a linear map
followed by a monomial map. Theorem 3.1 gives a combinatorial description of
the tropicalization of the image of such a map. The dual variety X∗ of any toric
A
variety X admits such a parametrization. This is derived in Section 4, and it
A
is used to prove Theorem 1.1 and Theorem 1.2 in the general form of Theorem
4.6. We also obtain in Corollary 4.5 a characterizationof the dimension of X∗. In
A
TROPICAL DISCRIMINANTS 3
Section5wediscusscomputationalissues,andweexaminethe connectionbetween
the tropical discriminant of A and regular polyhedral subdivisions. In particular,
weconsidertheproblemofcharacterizing∆-equivalenceofregulartriangulationsin
combinatorialterms. Finally,Section6isdevotedtothecasewhenAisanessential
Cayley configuration. The corresponding dual varieties X∗ are resultant varieties,
A
and we compute their degrees and initial cycles in terms of mixed subdivisions.
Acknowledgement: We thank the Forschungsinstitut fu¨r Mathematik at ETH
Zu¨rich for hosting Alicia Dickenstein and Bernd Sturmfels in the summer of 2005.
We are grateful to Jenia Tevelev and Josephine Yu for comments on this paper.
2. Tropical varieties and their initial cycles
Tropical algebraic geometry is algebraic geometry over the tropical semi-ring
(R∪{∞},⊕,⊙) with arithmetic operations x⊕y := min{x,y} and x⊙y := x+y.
Ittransferstheobjectsofclassicalalgebraicgeometryintothecombinatorialcontext
of polyhedral geometry. Fundamental references include [7, 18, 19, 20, 21, 24].
Tropicalizationisanoperationthatturnscomplexprojectivevarietiesintopoly-
hedral fans. Let Y ⊂ CPn−1 be an irreducible projective variety of dimension
r−1 and I ⊂ C[x ,...,x ] its homogeneous prime ideal. For w ∈ Rn, we de-
Y 1 n
note by in (I ) the initial ideal generated by all initial forms in (f), for f∈I ,
w Y w Y
whereinw(f)isthesubsumofalltermscuxu=cuxu11···xunn inf whichhavelowest
weight w·u= n w u . The tropicalization τ(Y) of Y is the set
i=1 i i
(2.1) τ(YP) = {w ∈Rn : in (I ) does not contain a monomial}.
w Y
Let K = C{{ǫR}} be the field of Puiseux series, i.e., series with complex coeffi-
cients,realexponentsandwellorderedsupports. TheelementsofK arealsoknown
as transfinite Puiseux series, and they form an algebraically closed field of charac-
teristic zero. The order of a non-zero element z in K∗ = K\{0} is the smallest
real number ν such that ǫν appears with non-zero coefficient in z. For a vector
z =(z ,...,z )in(K∗)s wewrite order(z):=(order(z ),...,order(z ))∈Rs. The
1 s 1 s
points in the tropicalization τ(Y) are precisely the orders of K∗-valued points on
the variety Y (see [7], [20, Theorem 2.1.2], [21, Theorem 2.1]).
The set τ(Y) carries the structure of a polyhedral fan. Namely, it is a subfan of
theGr¨obnerfanofI ;see[23,§9]. ByaresultofBieriandGroves[2],thefanτ(Y)
Y
is pure of dimension r. In [3] it was shown that τ(Y) is connected in codimension
one, and a practical algorithm was given for computing τ(Y) from polynomial
generators of I . We will view the tropicalization τ(Y) of a projective variety as
Y
an (r−1)-dimensional fan in tropical projective space TPn−1:=Rn/R(1,1,...,1),
which is an (n−1)-dimensional real affine space.
Every maximal cone σ of the fan τ(Y) comes naturally with an intrinsic multi-
plicity m , which is a positive integer. The integer m is computed as the sum of
σ σ
themultiplicities ofallmonomial-freeminimalassociatedprimesoftheinitialideal
in (I ) in C[x ,...,x ], where w is in the relative interior of the cone σ.
w Y 1 n
Remark 2.1. A geometric description of the intrinsic multiplicity m arises from
σ
thebeautifulinterplayofdegenerationsandcompactificationsdiscoveredbyTevelev
[24]andstudiedbySpeyer[20,Chapter2]andHacking(unpublished). LetXdenote
the toric variety associated with the fan τ(Y). Consider the intersection Y of Y
0
with the dense torusT inCPn−1, andlet Y be the closureofY in X. By [24, 1.7,
0 0
4 ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
2.5,and2.7],thevarietyY iscompleteandthemultiplicationmapΨ:T×Y →X
0 0
isfaithfullyflat. IffollowsthattheintersectionofY withacodimensionkorbithas
0
codimensionkinY . Inparticular,theorbitO(σ)correspondingtoamaximalcone
0
σ ofτ(Y) intersects Y in a zero-dimensionalscheme Z . The intrinsic multiplicity
0 σ
m of the maximal cone σ in the tropical variety τ(Y) is the length of Z .
σ σ
We list three fundamental examples which will be important for our work.
(1) Let Y be a hypersurface in CPn−1 defined by an irreducible polynomial f in
C[x ,...,x ]. Then τ(Y) is the union of all codimension one cones in the normal
1 n
fanofthe Newtonpolytope off. The intrinsic multiplicity m ofeachsuchcone σ
σ
is the lattice length of the corresponding edge of the Newton polytope of f.
(2) Let Y =X be the toric variety defined by an integer d×n-matrix A as
A
above. Its tropicalization τ(X ) is the linear space spanned by the rows of A.
A
(3) Let Y be a linear subspace in Cn or in CPn−1. The tropicalization τ(Y) is
the Bergmanfan of the matroid associated with Y; see [1, 10, 23] and (3.2) below.
In the last two families of examples, all the intrinsic multiplicities m equal 1.
σ
The tropicalization τ(Y) can be used to compute numerical invariants of Y.
First, the dimension of τ(Y) coincides with the dimension of Y. In Theorem 2.2
below, we express the multiplicities of the minimal primes in the initial monomial
ideals of I in terms of τ(Y). Equivalently, we compute the algebraic cycle of any
Y
initial monomial ideal in (I ). This formula tells us the degree of the variety Y,
w Y
namely,thedegreeisthesumofthemultiplicitiesoftheminimalprimesof in (I ).
w Y
Let c:=n−r denote the codimension of the irreducible projective variety Y in
CPn−1. Assume that Y is not contained in a coordinate hyperplane, and let I be
Y
its homogeneous prime ideal in C[x ,...,x ]. If w is a generic vector in Rn, the
1 n
initialidealin (I )isamonomialidealofcodimensionc. Everyminimalprimeover
w Y
in (I )isgeneratedbya subsetofc ofthe variables. We write P = hx : i∈τi
w Y τ i
forthemonomialprimeidealindexedbythesubsetτ ={τ ,...,τ }⊂{1,2,...,n}.
1 c
Assume that the cone w + R {e ,...,e } meets the tropicalization τ(Y).
>0 τ1 τc
We may suppose that the generic weight vector w ∈ Rn satisfies that the image
of w in TPn−1 does not lie in τ(Y) and that the intersection of the cone w +
R {e ,...,e } with τ(Y) is finite and contained in the union of the relative
>0 τ1 τc
interiorsofits maximalcones. Letσ be a maximalcone ofthe tropicalvariety and
(2.2) {v} = (L+w)∩L′,
where L=R{e ,...,e } and L′ =Rσ are the corresponding linear spaces, which
τ1 τc
are defined over Q. We associate with v the lattice multiplicity of the intersection
ofLandL′,whichisdefinedastheabsolutevalue ofthe determinantofanyn×n-
matrix whose columns consist of a Z-basis of Zn∩L and a Z-basis of Zn∩L′.
Here is the main result of this section.
Theorem 2.2. For w ∈Rn a generic weight vector, a prime ideal P is associated
τ
to the initial monomial ideal in (I ) if and only if the cone w+R {e ,...,e }
w Y >0 τ1 τc
meets the tropicalization τ(Y). The number of intersections, each counted with its
associated lattice multiplicity times the intrinsic multiplicity, is the multiplicity of
the monomial ideal in (I ) along the prime P .
w Y τ
Proof. We work over the Puiseux series field K = C{{ǫR}}, we write KPn−1 for
the (n−1)-dimensional projective space over the field K, and we consider Y as a
TROPICAL DISCRIMINANTS 5
subvariety of KPn−1. The translatedvariety ǫ−w·Y is defined by the prime ideal
ǫw·IY = f(ǫw1x1,...,ǫwnxn) : f ∈IY ⊂ K[x1,...,xn].
(cid:8) (cid:9)
Clearly, a point of the form w +u lies in τ(Y) if and only if there is a point in
ǫ−w· Y with order u.
Let L be a general linear subspace of dimension c in KPn−1 which is defined
over C. The intersection ǫ−w ·Y ∩ L is a finite set of reduced points in KPn−1.
Thenumberofthesepointsis thedegreedofY. Thereis aflatfamilyoverK with
generic fiber ǫ−w· Y and special fiber the scheme V(in (I )). Since L is generic,
w Y
theintersectionofLwiththisfamilyisstillflat. Then,allpointsintheintersection
ǫ−w · Y with L are liftings of points in V(in (I )) ∩ L (and all points can be
w Y
lifted). Moreover,the multiplicity of the P -primary component of the initial ideal
τ
in (I ) equals the number of points in ǫ−w· Y ∩ L of the form
w Y
θ·ǫu+... = θ ǫu1 +... : θ ǫu2 +... : ··· : θ ǫun +... ,
1 2 n
(cid:0) (cid:1)
where θ ∈C∗ for all k, u =0 for i6∈τ and u >0 for j ∈τ, i.e., the number of
k i j
pointswithvalues inthe intersectionofτ(Y)withthe cone w +R {e ,...,e }.
>0 τ1 τc
By our genericity assumption, each such intersection point v lies on some maxi-
mal cone σ of τ(Y) and it is counted with its multiplicity, which is the product of
the intrinsic multiplicity m times the lattice multiplicity of the transversal inter-
σ
sectionofrationallinearspacesin(2.2). Thisproductcanbeunderstoodbymeans
oftheflatfamilydiscussedinRemark2.1. Namely,itfollowsfromtheT-invariance
of the multiplication map Ψ that the scheme-theoretic fiber of Ψ over any point of
O(σ)isisomorphictoT′×Z ,whereT′isthestabilizerofapointinO(σ). SinceX
σ
is normal,T′ is a torus(C∗)r−1. By [24, 1.7], (C∗)r−1×Z is the intersectionofT
σ
withtheflatdegenerationofY inCPn−1 givenbytheoneparametersubgroupofT
specified by the rational vector w. Our construction above amounts to computing
the intersection of (C∗)r−1×Z with the corresponding degeneration of L. The
σ
lattice index is obtained from the factor (C∗)r−1, and m is obtained from the
σ
factor Z . Their product is the desired intersection number. (cid:3)
σ
ThealgebraiccycleofthevarietyY isrepresentedbyitsChow form Ch ,which
Y
isapolynomialinthebracketvariables[γ]=[γ ···γ ];see[13,§3.2.B].Theorem2.2
1 c
implies thatthe w-leadingtermofthe ChowformChY equals [γ]uγ,where uγ is
the (correctly counted) number of points in τ(Y) ∩ (w+R>0Q{eγ1,...,eγc}). We
discuss this statement for the three families of examples considered earlier.
(1) If Y is a hypersurface then c = 1 and the bracket variable [γ] is simply the
ordinaryvariablex fori=γ . Thew-leading monomialofthe defining irreducible
i 1
polynomialf(x1,...,xn)equalsxu11···xunn whereui isthenumberoftimestheray
w+R e intersects the tropical hypersurface τ(Y), counted with multiplicity.
>0 i
(2)IfY =X isatoricvarietythenτ(Y)=rowspace(A)andTheorem2.2implies
A
thefamiliarresult[13,Thm.8.3.3]thattheinitialcyclesofX aretheregulartrian-
A
gulationsofA. Indeed,w+R {e ,...,e }intersectstherowspaceofAifand
>0 γ1 γn−d
only if the (d−1)-simplex γ¯ ={a :i6∈γ} appearsin the regulartriangulationΠ
i w
of A induced by w. For a precise definition of Π see Section 5. The intersection
w
multiplicity is the lattice volume of γ¯. Hence in (Ch )= [γ]vol(γ¯).
w XA γ¯∈Πw
(3) If Y is a linear space then its ideal IY is generated by cQlinearly independent
linear forms in C[x ,...,x ]. The tropical variety τ(Y) is the Bergman fan, to be
1 n
discussedinSection3,andtheGr¨obnerfanofI isthenormalfanoftheassociated
Y
6 ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
matroid polytope [10]. For fixed generic w, there is a unique c-element subset γ
of[n]suchthatw+R {e ,...,e }intersectsτ(Y). Theintersectionmultiplicity
>0 γ1 γc
is one, and the corresponding initial ideal equals in (I )=hx ,...,x i.
w Y γ1 γc
3. Tropicalizing maps defined by monomials in linear forms
In this section we examine a class of rational varieties Y whose tropicalization
τ(Y)canbecomputedcombinatorially,withoutknowingtheidealI . Weconsider
Y
a rational map f : Cm 99K Cs that factors as a linear map Cm → Cr followed by
a Laurent monomial map Cr 99K Cs. The linear map is specified by a complex
r×m-matrix U =(u ), and the Laurent monomial map is specified by an integer
ij
s×r-matrixV =(v ). Thusthei-thcoordinateoftherationalmapf :Cm 99KCs
ij
equals the following monomial in linear forms:
r
(3.1) fi(x1,...,xm) = (uk1x1+···+ukmxm)vik, i=1,...,s.
kY=1
Let Y denote the Zariskiclosure of the image of f. Observe that if all row sums
UV
of V are equal then f induces a rational map CPm−1 99K CPs−1, and the closure
of its image is an irreducible projective variety, which we also denote by Y .
UV
Ourgoalistocomputethetropicalizationτ(Y )ofthevarietyY intermsof
UV UV
the matrices U and V. In particular, we will avoid any reference to the equations
of Y . The general framework of this section will be crucial for our proofs of the
UV
results on A-discriminants and their tropicalization stated in the Introduction.
We list a number of special cases of varieties which have the form Y .
UV
(1) If r=s, and V =I then f is the linear map x 7→ Ux, and Y is the image
r UV
ofU. WedenotethislinearsubspaceofCr byim(U). Itstropicalizationτ(im(U))
is the Bergman fan of the matrix U, to be discussed in detail below.
(2) If m=r and U=I then f is the monomial map specified by the matrix V.
m
Hence Y coincides with the toric variety X which is associated with the
UV Vt
transpose Vt of the matrix V. Its tropicalization is the column space of V.
(3) Let m=2, suppose the rows of U are linearly independent, and suppose the
matrix V has constant row sum. Then Y =image(CP1 →CPs−1) is a rational
UV
curve. Every rational projective curve arises from this construction, since every
binary form is a monomial in linear forms.
Our nexttheoremimplies thatτ(Y ) consists ofthe raysin TPs−1 spannedby
UV
the rows of V. Theorem 3.1 can also be derived from [24, Proposition 3.1], but we
prefer to give a proof that is entirely self-contained.
Theorem 3.1. The tropical variety τ(Y ) equals the image of the Bergman fan
UV
τ(im(U)) under the linear map Rr →Rs specified by the matrix V.
Proof. InwhatfollowsweconsiderallalgebraicvarietiesoverthefieldK =C{{ǫR}}
andweusethecharacterizationofthetropicalvarietyτ(Y )intermsofK∗-valued
UV
pointsofthe idealofY . Extendingscalars,weconsiderthemapf :Km 99KKs.
UV
Let z = f(x) be any point in the image of that map. For k ∈ {1,...,r} we set
y = u x +···+u x and α =order(y ). The vector y =(y ,...,y ) lies in
k k1 1 km m k k 1 r
the linear space im(U), and hence the vector α=(α ,...,α ) lies in τ(im(U)).
1 r
The order of the vector z =f(x)∈Ks is the vector V ·α∈Rs, since the order
r
of its ith coordinate z equals v ·α . Hence z =f(x) lies in V ·τ(im(U)),
i k=1 jk k
the image of the Bergman fan τP(im(U)) under the linear map V.
TROPICAL DISCRIMINANTS 7
Theimageoff isZariskidenseinY ,i.e.,thereexistsapropersubvarietyY of
UV
Y such that Y \Y contains the image of f. By the Bieri-Groves Theorem [2],
UV UV
τ(Y)isapolyhedralfanoflowerdimensioninsidethepure-dimensionalfanτ(Y ).
UV
Fromthis it follows that τ(Y ) is the closureof τ(Y \Y) in Rs. In the previous
UV UV
paragraph we showed that τ(Y \Y) is a subset of V ·τ(im(U)). Since the latter
UV
is closed, we conclude that τ(Y )⊆V ·τ(im(U)).
UV
It remains to show the converse inclusion V ·τ(im(U)) ⊆ τ(Y ). Take any
UV
pointβ ∈V ·τ(im(U)), andchooseα∈τ(im(U)) suchthat V ·α=β. There exists
a K∗-valued point y in the linear space im(U) such that order(y) = α. Then the
point yV = r yv1k,..., r yvsk lies in (K∗)s∩Y , and its order clearly
k=1 k k=1 k UV
equals β. He(cid:0)nQce β ∈τ(YUV)Qas desired(cid:1). (cid:3)
Remark 3.2. The intrinsic multiplicity m of any maximal cone σ in a sufficiently
σ
finefanstructureonthetropicalvarietyτ(Y )isalatticeindexwhichcanberead
UV
off from the matrix V. Namely, suppose σ lies the image of a maximal cone σ′ of
the Bergmanfan τ(im(U)). Then m is the index of the subgroup V(Rσ′∩Zr) of
σ
Rσ∩Zs. ThisfollowsfromRemark2.1usingstandardargumentsoftoricgeometry.
Theorem 3.1 and Remark 3.2 furnish a combinatorial construction for the trop-
icalization of any variety which is parameterized by monomials in linear forms.
Using the results of Section 2, Theorem 3.1 can now be applied to compute geo-
metric invariants of such a variety, for instance, its dimension, its degree and its
initialcycles. Tomakethiscomputationeffective,weneedanexplicitdescriptionof
the Bergmanfanτ(im(U)). Luckily, the relevantcombinatoricsis wellunderstood,
thanksto[1,10],andintheremainderofthissectionwesummarizewhatisknown.
LetM denotethematroidassociatedwiththerowsofther×m-matrixU. Thus
M is a matroid of rank at most m on the ground set [r]={1,2,...,r}. The bases
of M are the maximal subsets of [r] which index linearly independent rows of U.
Fix a vector w ∈ Rr. Then the w-weight of a basis β of M is w . Consider
i∈β i
the set of all bases of M that have maximal w-weight. This collPection is the set of
bases of a new matroid which we denote by M . Note that each M has the same
w w
rank and the same groundset as M =M . An element i of [r] is a loop of M if it
0 w
does not lie in any basis of M of maximal w-weight.
We can now describe τ(im(U)) in terms of the matroid M:
(3.2) τ(im(U)) = {w∈Rr : M has no loop}.
w
Thisrepresentationendowsourtropicallinearspacewiththestructureofapolyhe-
dralfan. Namely,ifw ∈τ(im(U)),thenthesetofallw′ ∈Rr suchthatMw′ =Mw
is a relatively open convex polyhedral cone in Rr. The collection of these cones is
denotedB(M)andis calledthe Bergman fan of the matroidM. Depending onthe
context, we may also write B(U) and call it the Bergman fan of the matrix U.
We now recall the connection between Bergman fans and nested set complexes.
The latter encode the structure of wonderful compactifications of hyperplane ar-
rangement complements in the work of De Concini and Procesi [5], and they were
later studied from a combinatorial point of view in [8, 9, 11].
A subset X ⊆ [r] is a flat of our matroid M if there exists a vector u ∈ im(U)
such that X = {i ∈ [r] : u =0}. The set of all flats, ordered by inclusion, is the
i
geometric lattice L=L . A flat X in L is called irreducible if the lower interval
M
{Y ∈L : Y ≤X} does not decompose as a direct product of posets. Denote by I
the set of irreducible elements in L. In other contexts, the irreducible elements of
8 ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
a lattice of flats of a matroid were named connected elements or dense edges. The
matroidM isconnectedifthetoprankflatˆ1=[r]isirreducible,andweassumethat
thisisthecase. Otherwise,weartificiallyaddˆ1toI. WecallasubsetS ⊆I nested
if for any set of pairwise incomparable elements X ,...,X in S, with t ≥ 2, the
1 t
join X ∨...∨X is not contained in I. The nested subsets in I form a simplicial
1 t
complex, the nested set complex N(L). See [8, Sect. 2.3] for further information.
FeichtnerandYuzvinsky[11,Eqn(13)]introducedthefollowingnaturalgeomet-
ric realization of the nested set complex N(L). Namely, the collection of cones
(3.3) R {e : X ∈S} for S ∈N(L)
≥0 X
formsaunimodularfanwhosefaceposetisthefaceposetofthenestedsetcomplex
ofL. Here e = e denotestheincidencevectorofaflatX ∈I. Weconsider
X i∈X i
this fan in the troPpical projective space TPn−1, and we also denote it by N(L).
Itwasshownin[10, Thm4.1]thatthenested set fan N(L )is asimplicialsub-
M
divisionoftheBergmanfanB(M),andhenceofourtropicallinearspaceτ(im(U)).
TheBergmanfanneednotbesimplicial,sothenestedsetfancanbefinerthanthe
Bergman fan. However,in many important cases the two fans coincide [10, §5].
What we defined above is the coarsest in a hierarchy of nested set complexes
associated with the geometric lattice L. Namely, for certain choices of subsets
G in L, the same construction gives a nested set complex N(L,G) which is also
realizedasaunimodularsimplicialfan. SuchG arecalledbuilding sets;thereisone
nested set fan for each building set G in L, see [8, Sect. 2.2, 2.3]. If two building
setsarecontainedinanother,G ⊆G ,thenN(L,G )isobtainedfromN(L,G )by
1 2 2 1
a sequence of stellar subdivisions [9, Thm 4.2]. The smallest building set is G =I,
thecasediscussedabove,andthelargestbuildingsetisthesetofallflats,G =L. In
the letter case, the correspondingnested set complex N(L,L) is the order complex
or flag complex F(L) of the lattice, i.e., the simplicial complex on L\{ˆ0} whose
simplices are the totally orderedsubsets. Again, there is a realizationof F(L) as a
unimodular fan with rays generated by the incidence vectors of flats in L. We call
this fan the flag fan of L and denote it by F(L) for ease of notation.
Summarizing the situation, we have the following sequence of subdivisions each
of which can be used to compute the tropicalization of a linear space.
Theorem 3.3. Given a matrix U and the matroid M of rows in U, the tropical
linear space τ(im(U)) has three natural fan structures: the Bergman fan B(M) is
refined by the nested set fan N(L ), which is refined by the flag fan F(L ).
M M
We present an example which illustrates the concepts developed in this section.
Example 3.4. Let m=4, r=5, s=4, and consider the map f=(f ,f ,f ,f ) from
1 2 3 4
C4 to C4 whose coordinates are the following monomials in linear forms:
f = (x −x )3(x −x )3
1 1 2 1 3
f = (x −x )2(x −x )2(x −x )2
2 1 2 2 3 2 4
f = (x −x )2(x −x )2(x −x )2
3 1 3 2 3 3 4
f = (x −x )3(x −x )3.
4 2 4 3 4
TROPICAL DISCRIMINANTS 9
So f is as in (3.1) with
1 −1 0 0
3 3 0 0 0
 1 0 −1 0 
2 0 2 2 0
U = 0 1 −1 0 and V = .
  0 2 2 0 2
 0 1 0 −1   
  0 0 0 3 3
 0 0 1 −1   
 
The projectivization of the variety Y =cl(image(f)) is a surface in CP3, since U
UV
has rank 3. We construct the irreducible homogeneous polynomial P(z ,z ,z ,z )
1 2 3 4
whichdefinesthis surface. ThematroidofU isthe graphicmatroidofK withone
4
edge removed. From [10, Example 3.4] we know that the Bergman complex B(U)
is the complete bipartite graph K , embedded as a 2-dimensional fan in TP4.
3,3
The tropical surface τ(Y ) is the image of B(U) under V. This image is a
UV
2-dimensional fan in TP3. It has seven rays: six of them are images of the rays
of B(U), the last one is the intersection of the images of two 2-dimensional faces
that occurs due to the non-planarity of K . Hence the Newton polytope of the
3,3
polynomialP is 3-dimensionalwith 6 vertices,11 edges,and 7 facets; see Figure 1.
The six extreme monomials of P can be computed (even by hand) using The-
orem 2.2, namely, by intersecting the rays w+R e with τ(Y ) in TP3. This
>0 i UV
computationrevealsin particular that the degreeof the polynomial P is 28. Using
linearalgebra,itisnoweasytodetermineall171monomialsintheexpansionofP.
z2z9z15z2 z12z12z4
1 2 3 4 2 3 4
z8z12z8 z8z12z8
1 3 4 1 2 4
z4z12z12 z2z15z9z2
1 2 3 1 2 3 4
Figure 1. The Newton polytope of the polynomial P in Example 3.4.
4. Back to A-discriminants
In this section we return to the setting of the Introduction, and we prove The-
orem 1.1 and Theorem 1.2. Recall that A is an integer d×n-matrix such that
(1,...,1) is in the row span of A, i.e., the column vectors a ,a ,...,a lie in an
1 2 n
affine hyperplane in Rd. We also assumed that the vectors a ,...,a span Zd. We
1 n
identify the matrix A with the point configuration {a ,a ,...,a }. The convex
1 2 n
hull of the configuration A is a (d−1)-dimensional polytope with ≤n vertices.
The projective toric variety X is defined as the closure of the image of the
A
monomial map ψA :(C∗)d →CPn−1, t7→(ta1 :ta2 : ··· : tan). Equivalently, XA
isthesetofallpointsx∈CPn−1 suchthatxu =xv forallu,v ∈Nn withAu=Av.
Let(CPn−1)∗denotetheprojectivespacedualtoCPn−1. Thepointξ =(ξ :···:
1
ξ ) in (CPn−1)∗ corresponds to the hyperplane H = {x ∈ CPn−1 : n x ξ =
n ξ i=1 i i
P
10 ALICIADICKENSTEIN,EVAMARIAFEICHTNER,ANDBERNDSTURMFELS
0}. The dual variety X∗ is definedas the closurein(CPn−1)∗ ofthe setofpoints ξ
A
such that the hyperplane H intersects the toric variety X at a regular point p
ξ A
and contains the tangent space T (p) of X at p.
XA A
Kapranov[16]showedthat reduceddiscriminantalvarietiesareparametrizedby
monomials in linear forms. This parametrization, called the Horn uniformization,
will allow us to determine the tropical discriminant τ(X∗) via the results of Sec-
A
tion 3. We denote by CP(ker(A)) the projectivization of the kernel of the linear
map given by A, an (n−d−1)-dimensional projective subspace of CPn−1, and we
denotebyTd−1 =(C∗)d/C∗ thedensetorusofX . Thefollowingresultisavariant
A
of [16, Theorem 2.1]; see also [13, §9.3.C].
Proposition 4.1. The dual variety X∗ of the toric variety X is the closure of
A A
the image of the map ϕ :CP(ker(A))×Td−1 →(CPn−1)∗ which is given by
A
(4.1) ϕA(u,t) = (u1ta1 : u2ta2 : ··· : untan).
Proof. Consider the unit point 1 = (1 : 1 : ... : 1) on the toric variety X . The
A
hyperplaneH containsboththepoint1andthetangentspaceT (1)atthispoint
ξ XA
if and only if ξ lies in the kernel of A. This follows by evaluating the derivative of
the parametrizationψ of X at (t ,t ,...,t )=(1,1,...,1). If p=ψ (t) is any
A A 1 2 d A
point in the dense torus of X , then the tangent space at that point is gotten by
A
translating the tangent space at 1 as follows:
T (p) = p·T (1).
XA XA
The hyperplane H contains p if and only if p−1·H = H contains 1, and H
ξ ξ ξ·p ξ
contains T (p) if and only if H contains T (1). These two conditions hold,
XA ξ·p XA
for some p in the dense torus of X , if and only if ξ ∈image(ϕ ). (cid:3)
A A
Proposition 4.1 shows that the dual variety X∗ of the toric variety X is
A A
parametrized by monomials in linear forms. In the notation of Section 3 we set
m=n, r =n+d, s=n, and the two matrices are
B 0
(4.2) U = and V = I At ,
(cid:18)0 Id(cid:19) n
(cid:0) (cid:1)
where B is an n × (n−d)-matrix whose columns span the kernel of A over the
integers. Thus the rows of B are Gale dual to the configuration A.
Lemma 4.2. The variety Y ⊂ Cn defined by (4.2) as in Section 3 is equal to
UV
the cone over the dual variety X∗ ⊂(CPn−1)∗ of the toric variety X ⊂CPn−1.
A A
Proof. Let f be the rational map defined by (4.2) as in Section 3, and set x =
(x ,...,x ) and t=(t ,...,t )=(x ,...,x ). Then (3.1) equals
1 n−d 1 d n−d+1 n
fi(x,t) = (bi,1x1+···+bi,n−dxn−d)·tai,
which equals the i-th coordinate of ϕ if we write ker(A) as the image of B. (cid:3)
A
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. Let us first note that the co-Bergman fan B∗(A) of the
rankdconfigurationgivenbythe columnsofAequalsthe Bergmanfanoftherank
n−d configuration given by the rows of B. Thus B∗(A), as it appears in (1.1) is
the Bergman fan of the matroid dual to the matroid given by the columns of A.