Definition 2.3.1. Let T = R ∪ {−∞}. The tropical semifield (T, ⊕, ⊙) is the semifield with operations a⊕ b := max{a, b} and a ⊙ b := a + b. (c.f.
Example 2.2.10)
Remark 2.3.2. Since (T\{−∞}, ⊙) is an abelian group, we can define the tropical division by
x� y := x − y,
for all x and y in T\{−∞}.
Proposition 2.3.3. (a) Both addition and multiplication are commutative:
x⊕ y = y ⊕ x and x ⊙ y = y ⊙ x. (b) The distributive law holds for tropical addition and tropical multiplication: x⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z.
Proof. (a) (i) x⊕ y = max{x, y} = max{y, x} = y ⊕ x.
(ii) x⊙ y = x + y = y + x = y ⊙ x.
(b) x ⊙ (y ⊕ z) = x + (y ⊕ z) = x + max{y, z} = max{x + y, x + z} = (x + y)⊕ (x + z) = x ⊙ y ⊕ x ⊙ z.
Remark 2.3.4. For all integer n and all x in T, we define
x⊙n:= x ⊙ · · · ⊙ x =
�n i=1
x = nx.
Note that−∞ is the additive identity and zero is the multiplicative unit, that is, x⊕ (−∞) = x and x ⊙ 0 = x.
Definition 2.3.5. The Rn is a module over the tropical semiring
(R∪ {∞}, ⊕, ⊙), with the operations of coordinatewise tropical addition
(a1,· · · , an)⊕ (b1,· · · , bn) = (max{a1, b1}, · · · , max{an, bn})
and tropical scalar multiplication
λ⊙ (a1,· · · , an) = (λ + a1,· · · , λ + an)
Definition 2.3.6. Let (M,⊕M) be a commutative monoid over tropical semi-field T. Then M is called a tropical module if there exists a scalar multipli-cation ⊙M : T× M → M denoted by ⊙M(t, m) = t⊙M x for all t in T and x in M , such that for all t1, t2 in T and x, y in M ,
(i) t1⊙M (x⊕M y) = (t1⊙M x)⊕M (t1 ⊙M y);
(ii) t1⊙M (t2⊙M x) = (t1⊙ t2)⊙M x;
(iii) 1T⊙M x = x where 1T= 0 is the multiplicative identity of T;
(iv) if t1⊙M x = t2⊙M x then either t1 = t2 or x =−∞.
For careful statements, we refer the reader to [21].
Definition 2.3.7. A T-vector space or tropical vector space M over T con-sists of a commutative monoid (M,⊕M) and ⊙M : T× M → M such that
for all t1, t2 in T, x, y in M , we have:
(i) t1⊙M (x⊕M y) = (t1⊙M x)⊕M (t1 ⊙M y);
(ii) t1⊙M (t2⊙M x) = (t1⊙ t2)⊙M x;
(iii) 1T⊙M x = x for the tropical multiplicative identity 1T.
(iv) (t1⊕ t2)⊙M x = (t1⊙M x)⊕M (t2 ⊙M x);
Definition 2.3.8. The tropical projective n-space, denoted by TPn, is de-fined as the quotient
(Tn+1\ (−∞, . . . , −∞))�
∼,
where ∼ denotes the equivalence relation, (x0, . . . , xn) ∼ (y0, . . . , yn) if and only if there exists a λ in T∗ such that (y0, . . . , yn) = (λ⊙ x0, . . . , λ⊙ xn) = (λ + x0, . . . , λ + xn).
Definition 2.3.9. Fix a weight vector ω = (ω1, . . . , ωn)∈ Rn. The weight of the variable xi is ωi. The weight of a term p(t)· xα11· · · xαnn is the real number
order(p(t)) + α1ω1+· · · + αnωn.
Definition 2.3.10. The tropical monomial is defined to be an expression of
the form
c⊙ xa11 ⊙ · · · ⊙ xann
where a1,· · · , an ∈ Z≥0 and c is a constant.
Definition 2.3.11. The finite linear combination of tropical monomials is called a tropical polynomial. Namely, f = c1⊙ xa111⊙ · · · ⊙ xan1n ⊕ · · · ⊕ ck⊙ xa1k1⊙ · · · ⊙ xankn
Definition 2.3.12. Consider a polynomial f ∈ C[x1, . . . , xn] and a vector ω ∈ Rn, the initial form inω(f ) is the sum of all terms in f of smallest ω-weight.
Definition 2.3.13. The tropical hypersurface of f is the set
T (f) = {ω ∈ Rn | inω(f ) is not a monomial}.
Remark 2.3.14. All of points ω of the T (f) are attained by at least two of the linear functions. Note that T (f) is invariant under dilation, so we can say T (f) by giving its intersection with the unit sphere. (See [2] and the references therein)
Chapter 3
Toric variety and Fano variety
We begin by recalling the some basic definitions and notations which are necessary for study tropical toric varieties.
3.1 Polyhedral Geometry
In this section, we will recall the polyhedral geometry since they relate to affine toric varieties and tropical toric varieties.
Definition 3.1.1. Let R be a ring. A right R-module M over R is an abelian group, usually written additively, and an operation M × R → M (denoted (m, r) �→ mr) such that for all r, s in R, x, y in M, we have:
(i) (x + y)r = xr + yr.
(ii) x(r + s) = xr + xs.
(iii) (xs)r = x(sr).
(iv) x1R= x if R has multiplicative identity 1R.
Similarly, we can define a left R-module via an operation R× M → M denoted (m, r) �→ rm and satisfy the above conditions. If R is a ring with identity, then a right R-module is also called a unitary right R-module. If R is a commutative ring, then a right R-modules are the same as left R-modules with mr = rm for all m in M , r in R and are called R-modules.
If R is a field, then a R-module M is called a vector space.
Definition 3.1.2. An abelian group F is called a free abelian group if it has a basis.
Example 3.1.3. The trivial group{0} is the free abelian group on the empty basis.
Definition 3.1.4. Let R be a ring. Let M be a right module and N be a left module over R. Let F be the free abelian group on M × N. Let K be the subgroup of F generated by all elements of the forms
(i) (a + b, c)− (a, c) − (b, c);
(ii) (a, c + d)− (a, c) − (a, d);
(iii) (ar, c)− (a, rc),
for all a, b ∈ M; c, d ∈ N; r ∈ R. The quotient group F/K is called the tensor product of M and N , and we write M⊗RN or simply M⊗N forF/K.
The element (a, c) in F/K is denoted by a⊗ c.
We denote by N � Zn the free abelian group and NR := N ⊗Z R the associated real vector space; moreover, we denote by M := hom(N, Z) the dual lattice of N and MR := M ⊗ZR.
Definition 3.1.5. The polyhedron P is the intersection of finitely many halfspaces in NR, that is, a set of the form
Figure 3.1: rational polyhedron
Definition 3.1.7. For every finite set S ⊆ Rd, if a set S is not convex set, the convex hull of S is the smallest convex set containing it, which we denote it by conv(S), that is,
conv(S) :=�
{K ⊆ Rd| S ⊆ K, K is a convex set}
.
Proposition 3.1.8. Let S be a finite subset of Rn. Then
conv(S) ={λ1x1+· · · + λmxm | x1, . . . , xm ∈ S, λi ≥ 0,
�m i=1
λi = 1}
Proof. For any finite set {x1, . . . , xm} ⊆ S, λi ≥ 0 with�m
i=1λi = 1.
We have λ1x1 +· · · + λmxm = (1 − λm)��m i=1
λixi
1−λm
�+ λmxm for λm <
1. Therefore, �m
Definition 3.1.9. For every finite set S in a real vector space, the positive hull or conical hull of S is denoted by pos(S) and is the set
pos(S) ={�
i∈I
λimi | {mi}i∈I ⊆ S, λi ≥ 0}.
Note that if S =∅, then pos(∅) = {0}.
Definition 3.1.10. The Minkowski sum of two sets X and Y in a vector space, defined by X + Y , is the set {x + y | x ∈ X, y ∈ Y }
Definition 3.1.11. A set σ is called a polyhedral cone (or simply a cone later) if
σ = pos(S) ={�
i∈I
λimi | {mi}i∈I ⊆ S, λi ≥ 0}
where S ⊆ NR is finite.
By the Minkowski-Weyl theorem for cones, the cone σ is a finitely gen-erated if and only if σ = {X ∈ NR| AX ≥ 0} where A ∈ (MR)d and b∈ Rd.
(i.e. σ is a polyhedron). For more details see [28] Section 1.3, [4] Theorem
Definition 3.1.14. Let τ and σ be nonempty polyhedra. τ is called a f acet of σ if τ is a face of σ and dim(τ ) + 1 = dim(σ) (denoted by τ ≺ σ), that is, a facet τ is a face of codimension 1.
Definition 3.1.15. A polyhedral cone σ is said a pointed cone if the origin is a face of σ. Otherwise, the polyhedral cone is called a blunt.
Example 3.1.16. In R, C1 = {x ∈ R | x ≥ 0} is a pointed cone, and C2 ={x ∈ R | x > 0} is a blunt.
Definition 3.1.17. A cone σ is called simplicial if it is generated by a linearly independent subset of the lattice N , that is, σ = pos(C) is called simplicial cone if C is linearly independent.
Definition 3.1.18. A simplicial cone σ is called unimodular if it is generated by a subset of a basis of the lattice N .
Example 3.1.19. Let N = Z(1, 0) ⊕ Z(0, 1). We consider the cone σ = pos{(1, 0), (3, 2)} in N. Then {(1, 0), (3, 2)} is linearly independent, but (2, 1) is not in Z≥0(1, 0)⊕ Z≥0(3, 2)}. Hence the cone σ is simplicial.
Example 3.1.20. Let N = Z(1, 0)⊕Z(0, 1). Given the cone σ = pos{(1, 0), (1, 1)}
in N . Then {(1, 0), (1, 1)} is a linearly independent set, and Z≥0(1, 0)⊕ Z≥0(1, 1)} can generate all of integer vectors in the cone σ. Hence the cone σ is unimodular.
Definition 3.1.21. The set P = conv(S) = {�
i∈Iλimi | {mi}i∈I ⊆ S, λi ≥ 0,�
i∈Iλi = 1} is said a polytope in NR where S⊆ NR is finite.
The P = conv(S) + pos(V ) for some finite sets S, V in NR if and only if
Figure 3.4: The tropical curve T rop(f )
Figure 3.5: The Newton sub-division of T rop(f )
Example 3.1.24. The tropicalisation of
f = t2 · x3 + x2y + xy2+ t2· y3+ x2+ 1t · xy + y2+ x + y + t2
is the tropical curve (as illustrated in Figure 3.4)
T rop(f ) = 2⊙ x⊙3⊕ x⊙2⊙ y ⊕ x ⊙ y⊙2⊕ 2 · y⊙3⊕ x⊙2⊕ x ⊙ y ⊙ (−1) ⊕ y⊙2
⊕ x ⊕ y ⊕ 2
= max{2 + 3x, 2x + y, x + 2y, 2 + 3y, 2x, x + y − 1, 2y, x, y, 2}
The vertices of the tropical curve are:
(2, 0), (1, 1), (1, 0), (0, 2), (0, 1), (0,−1), (−1, 0), (−1, −1), (−2, −2)
The Newton subdivision of the tropical curve T rop(f ) is
N ew(T rop(f )) = conv{(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (2, 1)}.
We illustrate the Newton subdivision of in Figure 3.5.
Definition 3.1.25. A polyhedral complex ∆ is a collection of polyhedra such that the following the two conditions are satisfied: if U ∈ ∆ and F is a face of U , then F ∈ ∆; if U, V ∈ ∆, then U ∩ V is a face of U and V .
The empty set is in the polyhedral complex ∆, i.e. a polyhedral complex
∆ contains empty face.
Definition 3.1.26. F is a polyhedral fan if F is a polyhedral complex and each σ in F is a cone.
Note that we consider the fan is collection of non-empty polyhedral cones in this paper.
Example 3.1.27. Suppose that τ1 = pos{(−1, 0)}, τ2 = pos{(1, 1)}, τ3 = pos{(0, −1)}, σ1 = pos{(−1, 0), (0, −1)}, σ2 = pos{(−1, 0), (1, 1)}, and σ3 = pos{(0, −1), (1, 1)} Let F = {(0, 0), τ1, τ2, τ3, σ1, σ2, σ3}}. Then (0, 0) is the face of other elements of F and (0, 0) = τ1∩τ2 = τ1∩τ3 = τ2∩τ3, τ1 = σ1∩σ2
is the face of σ1 and σ2, τ2 = σ2∩ σ3 is the face of σ2 and σ3, τ3 = σ1∩ σ3 is the face of σ1 and σ3. Hence F is a fan.
Figure 3.6: The fan F
Definition 3.1.28. Let F be a polyhedral complex. We define the following two notations:
• F(k) is a collection of k-dimendional polyhedra of F .
• |F | = �
U∈F
U is said the support of F .
Example 3.1.29. Recall from Example 3.1.27 that
F ={σ1, σ2, σ3, τ1, τ2, τ3, (0, 0)}.
F(0) ={(0, 0)}, F(1) ={τ1, τ2, τ3}, F(2) ={σ1, σ2, σ3}.
|F | = �
U∈F
U = σ1∪ σ2∪ σ3∪ τ1∪ τ2∪ τ3∪ (0, 0).
Definition 3.1.30. A polyhedral fan F is a rational fan if all cones in F are rational polyhedra.
Definition 3.1.31. A polyhedral fan F in a vector space NR is complete if the support of F is NR, i.e. |F | = NR.
Example 3.1.32. Recall from Example 3.1.27 that F ={σ1, σ2, σ3, τ1, τ2, τ3, (0, 0)}.
Then |F | = �
U∈F
U = σ1∪ σ2∪ σ3∪ τ1∪ τ2∪ τ3∪ (0, 0) = R2.
Definition 3.1.33. Let σ be a pointed rational cone in NR. The dual cone
σ∨ := {v ∈ MR | �v, u� ≥ 0, ∀u ∈ σ}.
Example 3.1.34. Let N = Ze1⊕ Ze2 where e1 = (1, 0) and e2 = (0, 1) are the standard basis vectors, and let σ = {0}. Then NR = N ⊗ R � R2, and M = Hom(N, Z) = Ze∨1 ⊕ Ze∨2, thus MR = M⊗ R � R2. Since v· 0 ≥ 0 for all v in MR, we have
σ∨ ={v ∈ MR | �v, 0� ≥ 0}
= pos{(1, 0), (−1, 0), (0, 1), (0, −1)}.
Definition 3.1.35. Let P be a polytope in NR. We define the dual polytope
P∨ := {v ∈ MR | �u, v� ≥ −1 for all u ∈ P }.
Theorem 3.1.36 (Farkas’ Theorem). Let σ be a polyhedral cone in NR,
then the dual cone σ∨ is a polyhedral cone in MR.
Proof. See [24] Corollary 22.3.1, [8] P.11 and [28] § 1.4.
Definition 3.1.37. A rational polyhedral cone is called strongly convex if it contains non-zero linear subspaces, namely, it does not contain line through the origin.
Proposition 3.1.38. Let σ lie in NR � Rn be a polyhedral cone. Then the following conditions are equivalent:
(i) σ is strongly convex;
(ii) {0} is a face of σ;
(iii) σ∩ (−σ) = {0};
(iv) n is the dimension of σ∨;
(v) σ contains no positive-dimensional subspace of NR.
Lemma 3.1.39 (Separation Lemma). Let ∆ be a fan in NR, and let σ1 and σ2 be polyhedral cones in ∆. Let τ = σ1∩ σ2 be a common face of σ1 and σ2. Then there exists u in σ∨1 ∩ σ2∨ such that
τ = σ1 ∩ u⊥ = σ2∩ u⊥.
Proposition 3.1.40. Let σ be a unimodular cone. Let the set {u1, . . . , un}
Theorem 3.1.41 (Duality Theorem). If σ is a convex polyhedral cone in NR, then (σ∨)∨ = σ.
Proof. This is well known result. For careful information see [12] P.47 and
[24] Theorem 14.1.
Proposition 3.1.42. Let the set {u1, . . . , un} be a basis of N. Let σ = pos{u1, . . . , uk} be a unimodular cone in NR, then (σ∨)∨ = σ.
Proof. According to the above Proposition 3.1.40, we have
σ∨ = pos{u∨1, . . . , u∨k,±u∨k+1, . . . ,±u∨n}
since σ be a unimodular cone. So we get that
σ∨∨={ω ∈ NR|< ω, v >≥ 0, ∀v ∈ σ∨}.
Lemma 3.1.43 (Gordon’s Lemma). Let σ be a rational convex polyhedral
cone in NR, then Sσ := σ∨∩ M is a finitely generated semigroup where M is a daul lattice of N .
Proof. By the Farkas’ Theorem, the dual cone σ∨ is a polyhedral cone in MR � Rn. Let σ∨ = pos{U} where U = {u1, . . . , um} is a finite subset of
i=1tiui is in K∩ M. Therefore, u is a nonnegative integer combination of elements of U ∪ (K ∩ M).
The Gordon’s lemma is well known result. For more information also see [8], [22] or [28].
Example 3.1.44. Let N=Z(−1, 0) ⊕ Z(0, −1). Take σ = pos{(−1, 0)}. If (x1, x2)· (−1, 0) = −x1 ≥ 0, then the dual cone
σ∨ ={v ∈ MR | v · (−1, 0) ≥ 0}
= pos{(−1, 0), (0, 1), (0, −1)}
So the corresponding semigroup Sσ = σ∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(0, 1)⊕ Z≥0(0,−1).
Proposition 3.1.45. Let ∆ be a fan in NR, and let τ is a face of σ in ∆, then
Sτ = Sσ + Z≥0(−u)
for some −u in the dual lattice M.
Proposition 3.1.46. Take a fan ∆ in NR. Let σ1 and σ2 in ∆, and let τ = σ1∩ σ2, then
Sτ = Sσ1 + Sσ2