Fano varieties

In this section, we will introduce and outline the Fano variety and Fano polytope. For more information see [5], [22], [7], [8] and [14].

Definition 3.4.1. For all r in Z, we define the Hirzebruch surface

Hr:= {([x⁰ : x1], [y0 : y1 : y2])∈ CP¹× CP² | x^r0y0 = x^r₁y1}.

Since Hr is isomorphic to H_−r for all r in Z, we sometimes assume Z_≥0.

Theorem 3.4.2. Let P ba a polytope in NR. If all of facets of P are the convex hull of a basis of N if and only if XP is a smooth Fano variety.

Proof. See [14] Proposition 3.6.7 and [7] Lemma 8.5.

The Fano varieties in two-dimension are also called a del Pezzo surface.

Theorem 3.4.3. There exist five distinct toric Fano varieties of two-dimension up to isomorphism,

1. CP²,

2. CP¹× CP¹,

3. the equivariant blowing-up of CP² at one point (i.e. the Hirzebruch surface H₁),

4. the equivariant blowing-up of CP² at two point, 5. the equivariant blowing-up of CP² at three point.

Proof. See [22] Propsition 2.21.

Chapter 4

Tropical Toric Variety

4.1 K(G, R, M )

Definition 4.1.1. For all real number x, we define

x⁺ := max(x, 0),

x⁻ := max(−x, 0),

and called positive part and negative part of x, respectively.

Remark 4.1.2. The x⁺ and x⁻ are non-negative and x = x⁺− x⁻.

Definition 4.1.3. For all extended real-valued function f , the positive part

of f is defined by f⁺(x) := max{f(x), 0}, and the negaive part of f is defined by f⁻(x) := max{−f(x), 0}. So we have f = f⁺− f⁻.

Remark 4.1.4. Let f := (f1,· · · , fⁿ) ∈ R^1×n, then f⁺ = (f₁⁺,· · · , fn⁺), f⁻ = (f₁⁻,· · · , fn⁻), and f = f⁺− f⁻.

Proposition 4.1.5. Let f := (f1,· · · , fn) ∈ R^1×n and x := (x1,· · · , xn) ∈ Rⁿ, then the equations f · x = 0 if and only if f⁺· x = f⁻· x.

Proof. Suppose that f · x = 0 where f := (f¹,· · · , fⁿ) ∈ R^1×n and x :=

(x1,· · · , xⁿ) ∈ Rⁿ. Then (f1,· · · , fⁿ)· (x¹,· · · , xⁿ) = 0 implies f1 × x¹ +

· · · fn× xn = 0. Since fi = f_i⁺− fi⁻ for all i = 1,· · · , n, we have (f1⁺− f₁⁻)× x¹+· · · + (fn⁺− fn⁻)× xⁿ= 0. Therefore, f₁⁺× x¹+· · · + fn⁺× xⁿ= f₁⁻× x¹ +· · · + fn⁻× xⁿ. Hence f⁺· x = f⁻· x.

Conversely, assume that f⁺ · x = f⁻ · x where f⁺, f⁻ ∈ R^1×n and x :=

(x1,· · · , xⁿ)∈ Rⁿ, then f₁⁺× x¹+· · · + fn⁺× xⁿ= f₁⁻× x¹+· · · + fn⁻× xⁿ. So we have (f₁⁺− f1⁻)× x¹+· · · + (fn⁺− fn⁻)× xⁿ = 0. Hence f· x = 0 since fi = f_i⁺− fi⁻ for all i = 1,· · · , n and x := (x1,· · · , xn)∈ Rⁿ.

Definition 4.1.6. Let S be a semigroup in Zⁿ and let G = {g1,· · · , gm} be a finite set of generators of S. Let R = {r¹,· · · , r^k} ⊆ Z^m generate the integer relation between a set of G, that is, Span^Z(R) = {z ∈ Z^m | g1z1 +· · · + g^mzm = 0}. Let M be another commutative semigroup. We

define

K(G, R, M ) := �

x∈ M^|G|| r⁺· x = r⁻· x ∀r ∈ R�

We will discuss about that r· x = 0 is different from r⁺· x = r⁻· x in the tropical semifield T.

Example 4.1.7. Let S ⊆ Z² and let G = {(1, 1), (4, 4)} be the generating set of S. Then |G| = 2, and we have (1, 1)z¹+ (4, 4)z2 = 0 for all z1, z2 ∈ Z.

This implies R = {(−4, 1)} and Span^Z(R) = {(z1, z2) ∈ Z² | z1 + 4z2 = 0} Let M = T and let x = (x1, x2) ∈ T². Then r · x = 0^T where 0^T is the tropical additive identity. This implies (−4 ⊙ x¹)⊕ (1 ⊙ x²) = 0T, so max{−4 + x1, 1 + x2} = −∞. Hence (x1, x2) = (−∞, −∞).

However, r⁺· x = (0, 1) · (x1, x2) = (0⊙ x1)⊕ (1 ⊙ x2) = max{0 + x1, 1 + x2}.

Similarly r⁻·x = max{4+x¹, 0+x2}. So {(x¹, x2)∈ T² | r⁺· x = r⁻· x ∀r ∈ R} = {−∞, x1+ 3 = x2}. Then K(G, R, T) = {−∞, x1+ 3 = x2}

Hence r· x = 0 is different from r⁺· x = r⁻· x in T.

Proposition 4.1.8. Given G = {g¹, . . . , gm}. Let R = {r¹,· · · , r^k} ⊆ Z^m Let M be a tropical semifield T. Suppose that K = K(G, R, T) = {x ∈ T^m | r⁺· x = r⁻· x ∀r ∈ R}. We define two operations ⊕K : K× K → K and ⊗^K : T× K → K by x ⊕^Ky = (x1⊕ y¹, . . . , xm⊕ y^m) and t⊗^Kx = (t⊙x1, . . . , t⊙xm), respectively. Then (K,⊕K,⊗K) is a tropical vector space.

Proof. Let x = (x1, . . . , xm), y = (y1, . . . , ym), and v = (v1, . . . , vm) be in K, and let t, t1, and t2 be in T. Given r = (z1, . . . , zm) is in R. Then r⁺ · x = r⁻ · x and r⁺ · y = r⁻ · y. We claim that K is closed under an operations ⊕K.

r⁺· (x ⊕Ky) = (z₁⁺, . . . , z_m⁺)· (x1 ⊕ y1, . . . , xm⊕ ym)

= (z₁⁺⊙ (x1⊕ y1))⊕ · · · ⊕ (zm⁺⊙ (xm⊕ ym))

= (z₁⁺+ max{x1, y1}) ⊕ · · · ⊕ (zm⁺+ max{xm, ym})

= max{z1⁺+ x₁, z₁⁺+ y₁} ⊕ · · · ⊕ max{zm⁺+ x_m, z_m⁺+ y_m}

= ((z₁⁺⊙ x¹)⊕ (z1⁺⊙ y¹))⊕ · · · ⊕ ((zm⁺⊙ x^m)⊕ (zm⁺⊙ y^m))

= ((z₁⁺⊙ x¹)⊕ · · · ⊕ (zm⁺⊙ x^m))⊕ ((z1⁺⊙ y¹)⊕ · · · ⊕ (z⁺m⊙ y^m)) (since (T,⊕) is a commutative monoid.)

= ((z₁⁺, . . . , z_m⁺)· (x1, . . . , xm))⊕ ((z1⁺, . . . , z_m⁺)· (y1, . . . , ym))

= (r⁺· x) ⊕ (r⁺· y)

= (r⁻· x) ⊕ (r⁻· y)

= ((z₁⁻, . . . , z_m⁻)· (x¹, . . . , xm))⊕ ((z1⁻, . . . , z_m⁻)· (y¹, . . . , ym))

= ((z₁⁻⊙ x1)⊕ · · · ⊕ (zm⁻⊙ xm))⊕ ((z1⁻⊙ y1)⊕ · · · ⊕ (z⁻m⊙ ym))

= ((z₁⁻⊙ x1)⊕ (z1⁻⊙ y1))⊕ · · · ⊕ ((zm⁻⊙ xm)⊕ (zm⁻⊙ ym)) (since (T,⊕) is a commutative monoid.)

= max{z1⁻+ x1, z₁⁻+ y1} ⊕ · · · ⊕ max{zm⁻+ xm, z_m⁻+ ym}

= (z₁⁻+ max{x1, y1}) ⊕ · · · ⊕ (zm⁻+ max{xm, ym})

= (z₁⁻⊙ (x1⊕ y1))⊕ · · · ⊕ (zm⁻⊙ (xm⊕ ym))

= (z₁⁻, . . . , z_m⁻)· (x1 ⊕ y1, . . . , xm⊕ ym)

= r⁻· (x ⊕Ky)

Hence x⊕K y is in K.

We claim that K is closed under an operations ⊗K.

r⁺· (t ⊗^K x) = (z₁⁺, . . . , z_m⁺)· (t ⊙ x¹, . . . , t⊙ x^m)

= (z₁⁺⊙ (t ⊙ x1))⊕ · · · ⊕ (zm⁺⊙ (t ⊙ xm))

= ((z₁⁺⊙ t) ⊙ x1)⊕ · · · ⊕ ((zm⁺⊙ t) ⊙ xm) (since (T,⊙) satisfies associative.)

= ((t⊙ z1⁺)⊙ x1)⊕ · · · ⊕ ((t ⊙ zm⁺)⊙ xm) (since (T\ {0^T}, ⊙) is abelian.)

= (t⊙ (z1⁺⊙ x1))⊕ · · · ⊕ (t ⊙ (zm⁺⊙ xm))

= t⊙ ((z1⁺⊙ x¹)⊕ · · · ⊕ (zm⁺⊙ x^m)) ( since (T,⊕, ⊙) is a semifield.)

= t⊙ (r⁺· x)

= t⊙ (r⁻· x)

= t⊙ ((z1⁻⊙ x1)⊕ · · · ⊕ (zm⁻⊙ xm))

= (t⊙ (z1⁻⊙ x1))⊕ · · · ⊕ (t ⊙ (zm⁻⊙ xm)) (since (T,⊕, ⊙) is a semifield.)

= ((t⊙ z1⁻)⊙ x¹)⊕ · · · ⊕ ((t ⊙ zm⁻)⊙ x^m) (since (T\ {0^T}, ⊙) is abelian.)

= ((z₁⁻⊙ t) ⊙ x1)⊕ · · · ⊕ ((zm⁻⊙ t) ⊙ xm)

= (z₁⁻⊙ (t ⊙ x1))⊕ · · · ⊕ (zm⁻⊙ (t ⊙ xm)) (since (T,⊙) satisfies associative.)

= (z₁⁻, . . . , z_m⁻)· (t ⊙ x1, . . . , t⊙ xm)

= r⁻· (t ⊗Kx)

Hence t⊗Kx is in K.

x⊕K(y⊕Kv) = x⊕K(y1⊕ v1, . . . , ym⊕ vm)

= (x1⊕ (y1⊕ v1), . . . , xm⊕ (ym⊕ vm))

= ((x₁⊕ y1)⊕ v1), . . . , (x_m⊕ ym)⊕ vm)

= (x1⊕ y¹, . . . , xm⊕ y^m)⊕^Kv

= (x⊕^K y)⊕^Kv

(0^T, . . . , 0^T)⊕^Kx = (−∞, . . . , −∞) ⊕^K (x1, . . . , xm)

= ((−∞) ⊕ x1, . . . , (−∞) ⊕ xm)

= (x1, . . . , xm)

(t1⊙ t²)⊗^Kx = ((t1 ⊙ t²)⊙ x¹, . . . , (t1⊙ t²)⊙ x^m)

= (t1⊙ (t2⊙ x1), . . . , t1(⊙t2⊙ xm))

= t1⊗K(t2⊙ x1, . . . , t2⊙ xm)

= t1⊗K(t2⊗K x)

(t1⊕ t2)⊗Kx = ((t1⊕ t2)⊙ x1, . . . , (t1⊕ t2)⊙ xm)

= ((t₁⊙ x1)⊕ (t2⊙ x1), . . . , (t₁⊙ xm)⊕ (t2⊙ xm))

= (t1⊙ x¹, . . . , t1⊙ x^m)⊕^K(t2⊙ x², . . . , t1⊙ x^m)

= (t1⊗^Kx)⊕^K(t2⊗^Kx)

t⊗^K(x⊕^Ky) = t⊗^K(x1⊕ y¹, . . . , xm⊕ y^m)

= (t⊙ (x¹⊕ y¹), . . . , t⊙ (x^m⊕ y^m))

= ((t⊙ x1)⊕ (t ⊙ y1), . . . , (t⊙ xm)⊕ (t ⊙ ym))

= (t⊙ x1, . . . , t⊕ xm)⊕K(t⊙ y1, . . . , t⊕ ym)

= (t⊗Kx)⊕K(t⊗Ky)

Hence (K,⊕K,⊗K) is a tropical vector space.

Proposition 4.1.9. If M is a abelian group, then K(G, R, M ) is a abelian

group.

Proof. Let M be an abelian group with an operation ∗ : M × M → M by ∗(a, b) = a ∗ b. Then M^|G| is also an abelian group. Because G is finite set, so we can consider |G| = k. We want to show that K(G, R, M) is a subgroup of M^k. Let x = (x₁, . . . , x_k) and y = (y₁, . . . , y_k) in K(G, R, M ).

We claim that x∗ y⁻¹ = (x1∗ y⁻¹1 , . . . , xk∗ yk⁻¹) in K(G, R, M ) where y⁻¹ is the inverse for y. Since M is an abelian group, we have xi∗ y⁻¹i in M for all i = 1, . . . , k, thus x∗ y⁻¹ in M^k. Let r = (z₁, . . . , z_k) in SpanZR. Since M is an abelian, M is a Z-module, thus (z_i⁺· (xi∗ y⁻¹i )) = ((z⁺_i · xi)∗ (z⁺i · yi⁻¹)) and (z_i⁻· (xⁱ∗ yi⁻¹)) = ((z_i⁻· xⁱ)∗ (z⁻i · yi⁻¹)) for all i = 1, . . . , k. Moreover, r⁺· x = r⁻ · x and r⁺· y⁻¹ = r⁻· y⁻¹, because x = (x₁, . . . , x_k) and y = (y1, . . . , yk) in K(G, R, M ). So we have

r⁺· (x ∗ y⁻¹) = (z₁⁺, . . . , z_k⁺)· (x¹∗ y⁻¹1 , . . . , xk∗ y⁻¹k )

= (z₁⁺· (x1∗ y⁻¹1 ))∗ · · · ∗ (zk⁺· (xk∗ yk⁻¹))

= ((z₁⁺· x1)∗ (z1⁺· y1⁻¹))∗ · · · ∗ ((zk⁺· xk)∗ (z⁺k · y⁻¹k ))

= ((z₁⁺· x1)∗ · · · ∗ (zk⁺· xk))∗ ((z1⁺· y1⁻¹)∗ · · · ∗ (z⁺k · yk⁻¹))

= r⁺· x ∗ r⁺· y⁻¹

= r⁻· x ∗ r⁻· y⁻¹

= ((z₁⁻· x¹)∗ · · · ∗ (zk⁻· x^k))∗ ((z1⁻· y1⁻¹)∗ · · · ∗ (z⁻k · yk⁻¹))

= ((z₁⁻· x1)∗ (z1⁻· y1⁻¹))∗ · · · ∗ ((zk⁻· xk)∗ (z⁻k · y⁻¹k ))

= (z₁⁻· (x1∗ y⁻¹1 ))∗ · · · ∗ (zk⁻· (xk∗ yk⁻¹))

= (z₁⁻, . . . , z_k⁺)· (x1∗ y⁻¹1 , . . . , xk∗ y⁻¹k )

= r⁻· (x ∗ y⁻¹).

Hence we get x∗ y⁻¹ in K(G, R, M ), i.e. K(G, R, M ) is a group.

Next we claim that K(G, R, M ) is an abelian. Suppose that x = (x1, . . . , xk) and y = (y1, . . . , yk) in K(G, R, M ). Since M is an abelian, xi ∗ yⁱ = yi∗ xⁱ for all i = 1, . . . , k. Then

x∗ y = (x1, . . . , xk)∗ (y1, . . . , yk)

= (x1∗ y1, . . . , xk∗ yk)

= (y₁∗ x1, . . . , y_k∗ xk)

= y∗ x

Hence K(G, R, M ) is an abelian group.

By the above proposition, K(G, R, T\ {−∞}) is an abelian group since (T\ {−∞}, ⊙) is an abelian group.

Example 4.1.10. Let S ⊆ Z² and let G ={(1, 2), (2, 4)} be the generating set of S. Then |G| = 2, and we have (1, 2)z¹+ (2, 4)z2 = 0 for all z1, z2 ∈ Z.

This implies R ={r = (−2, 1)}. So r⁺= (0, 1) and r⁻ = (2, 0).

Given an abelian group M = (Z4, +). Let x = (x1, x2)∈ Z4× Z4. If r⁺· x = r⁻· x, then x² = 2x1, and so K(G, R, Z4) = {(x¹, x2)∈ Z⁴× Z⁴ | x² = x1} is subset of Z₄× Z4.

We claim that K(G, R, Z4) is a subgroup of Z²₄. Suppose that (x1, x2) and (y1, y2) are in K(G, R, Z4), then x2 = 2x1and y2 = 2y1. Then 2(x1+(−y¹)) = 2x₁ + 2(−y1) = 2x₁+ (−2y1) = x₂ + (−y2) where (−y1,−y2) is the inverse element of (y1, y2), so (x1+ (−y1), x2+ (−y2)) is in K(G, R, Z4).

Since xi+ yi = yi+ xi for all i = 1, 2, (x1, x2) + (y1, y2) = (x1+ y1, x2+ y2) = (y₁+ x₁, y₂+ x₂) = (y₁, y₂) + (x₁, x₂)

Hence K(G, R, Z4) is abelian group.

Proposition 4.1.11. If M is a ring, then K(G, R, M ) is a M -module.

Proof. Suppose that M is a ring with binary operation ∗, together with a second binary operation ⊗. Because (M, ∗, ⊗) is a ring, so (M, ∗) is an abelian group, thus K(G, R, M ) is also an abelian group.

Define a operation � : K(G, R, M) × M → K(G, R, M) via x � m = (x1⊗ m, . . . , xk ⊗ m). To check that it is well-defined. Suppose that r is in R and |G| = k. Let m, n be in M and let x = (x1, . . . , x_k), y = (y₁, . . . , y_k) in

K(G, R, M ). Then r⁺· x = r⁻· x.

r⁺· (x � m) = (z1⁺, . . . , z⁺_k)· (x1⊗ m, . . . , xk⊗ m)

= z₁⁺⊗ (x1⊗ m) ∗ · · · ∗ zk⁺⊗ (xk⊗ m)

= (z₁⁺⊗ x1)⊗ m ∗ · · · ∗ (zk⁺⊗ xk)⊗ m

= (z₁⁺⊗ x1, . . . , z⁺_k ⊗ xk)� m

= (r⁺· x) � m

= (r⁻· x) � m

= (z₁⁻⊗ x1, . . . , z⁻_k ⊗ xk)� m

= (z₁⁻⊗ x1)⊗ m ∗ · · · ∗ (zk⁻⊗ xk)⊗ m

= z₁⁻⊗ (x1⊗ m) ∗ · · · ∗ zk⁻⊗ (xk⊗ m)

= (z₁⁻, . . . , z⁻_k)· (x¹⊗ m, . . . , x^k⊗ m)

= r⁻· (x � m)

So x� m is in K(G, R, M).

Suppose that x = y (i.e. xi = yi for all i = 1, . . . , k) and m = n, then x� m = (x¹ ⊗ m, . . . , x^k⊗ m) = (y¹ ⊗ n, . . . , y^k⊗ n) = y � n. Hence it is well-defined.

(x∗ y) � m = (x1∗ y1, . . . , x_k∗ yk)� m

= ((x1∗ y¹)⊗ m, . . . , (x^k∗ y^k)⊗ m)

= ((x1⊗ m) ∗ (y1⊗ m), . . . , (xk⊗ m) ∗ (yk⊗ m))

= (x1⊗ m, . . . , xk⊗ m) ∗ (y1⊗ m, . . . , yk⊗ m)

= (x� m) ∗ (y � m)

x� (m ∗ n) = (x1, . . . , xk)� (m ∗ n)

= (x₁⊗ (m ∗ n), . . . , xk⊗ (m ∗ n))

= ((x1⊗ m) ∗ (x¹⊗ n), . . . , (x^k⊗ m) ∗ (x^k⊗ n))

= (x1⊗ m, . . . , x^k⊗ m) ∗ (x¹⊗ n, . . . , x^k⊗ n)

= (x� m) ∗ (x � n)

(x� m) � n = (x¹⊗ m, . . . , x^k⊗ m) � n

= ((x1⊗ m) ⊗ n, . . . , (xk⊗ m) ⊗ n)

= (x1⊗ (m ⊗ n), . . . , xk⊗ (m ⊗ n))

= x� (m ⊗ n)

If M has an identity 1M, that is m ⊗ 1M = m for all m ∈ M. Then x� 1^M = (x1⊗ 1^M, . . . , xk⊗ 1^M) = (x1, . . . , xk) = x.

Remark 4.1.12. If M is a field, then K(G, R, M ) is a vector space.

Example 4.1.13. Let M = GF (4) be a Galois field. Let S ⊆ Z² and let G = {(1, 2), (4, 8)} be the generating set of S. Then |G| = 2, and we have (1, 2)z1+ (4, 8)z2 = 0 for all z1, z2 ∈ Z. This implies R = {r = (−4, 1)}. So r⁺= (0, 1) and r⁻ = (4, 0). If r⁺· x = r⁻· x where x = (x¹, x2) is in GF (4)², then x2 = 4x1 = 0, and so K(G, R, GF (4)) ={(x1, x2)∈ GF (4)² | x2 = 0} is a subset of GF (4)². We claim that K(G, R, GF (4)) is a vector space, in fact, we just need to show that K(G, R, GF (4)) is a subspace of GF (4)². To start with, it is clearly that (0, 0) is in K(G, R, GF (4)). Next, suppose that x = (x1, x2) and y = (y1, y2) are in K(G, R, GF (4)), then x2 = 0 and y2 = 0, then x₂+y₂ = 0, and so x+y is in K(G, R, GF (4)). Finally, let c be in GF (4), and let x = (x1, x2) be in K(G, R, GF (4)), then x2 = 0 and cx = (cx1, cx2), then cx2 = 0, and so cx is in K(G, R, GF (4)). Hence K(G, R, GF (4)) is a subspace of GF (4)².

Note that the Galois field GF (4) is isomorphic to GF (2)�

(x²+ x + 1), in fact GF (2) � Z2 since GF (p)� Zp for all prime p.

Theorem 4.1.14. Let S be a finitely generated semigroup on Zⁿ. Let G = {g1,· · · , gl} be a set of generaters with relations generated by R = {r¹,· · · , r^k}. Let M be an additive semigroup. Then there is a bijection between hom(S, M ) and K(G, R, M ).

Proof. We define a map φ : hom(S, M ) → K(G, R, M). Define a function

K(G, R, M ). To check that φ is surjective, and ψ is injective, i.e. φ ◦ ψ identity function.

φ◦ ψ(x) = φ(ψ(x)) = (f(g¹), . . . , f (gl)) = (x1, . . . , xl) = x.

Proposition 4.1.15. Let S be a finitely generated semigroup on Zⁿ. Let G = {g1,· · · , gl} be a set of generaters with relations generated by R = {r¹,· · · , r^k}. Then there are bijective between Spec(C[S]), hom(S, C) and K(G, R, C).

Proof. The correspondence between Spec(C[S_σ]) and hom(S, C) is imme-diate from Proposition 3.3.11. The correspondence between hom(S, C) and K(G, R, C) is immediate from the Theorem 4.1.14.

Theorem 4.1.16. Let S be a finitely generated semigroup on Zⁿ. Let G = {g1,· · · , gl} and H = {h1,· · · , hd} be the different sets of the generators of S with relations generated by R = {r¹,· · · , r^k}and P = {p¹,· · · , p^m}, re-spectively. Then there is a linear isomorphism φ : K(G, R, R)→ K(H, P, R).

Proof. Since G = {g1,· · · , gl} and H = {h1,· · · , hd} are the different sets of the generators of S, let gi = �d

j=1λijhj with nonnegative integer λij for all i = 1, . . . , l, and let hi = �l

i=1µjigi with nonnegative integer µji for all j = 1, . . . , d. Since hom(S, R) is in bijection with K(G, R, R), let x_i = f (g_i)

for all i = 1, . . . , l where f is in hom(S, R). Then �l

=

i=1µdixi). Then the function is well-defined by the above discussion.

−x1}, thus S = {(1, −2, −1}. So we have

K(H, P, R) ={(x1, x2, x3)∈ R³ | (1, 0, 0) · (x1, x2, x3) = (0, 2, 1)· (x1, x2, x3)}

={(x1, x2, x3)∈ R³ | x1 = 2x2+ x3}.

Then the function φ : K(G, R, R) → K(H, P, R) via φ(x1, x2) = (x1 + 2x2, x2, x1) is a linear isomophism.

4.2 Tropical Toric Variety

Definition 4.2.1. Let T be a tropical semifield. Let σ be a rational poly-hedral cone with the semigroup Sσ. Then the affine toric variety is Uσ :=

hom(S_σ, T) where hom(S_σ, T) is the semigroup homomorphisms S_σ → T.

Note that we set T^∗ = T\ {−∞}.

Example 4.2.2. Let N � Z² be a lattice with associated vector space N^R � R², and let M be a dual lattice of N with associated vector space MR � R². Given a cone σ = pos{0} in N^R, then its dual cone {0}^∨ = pos{(1, 0), (−1, 0), (0, 1), (0, −1)}, then S{0}={0}^∨∩M = Z≥0(1, 0)⊕Z≥0(−1, 0)⊕

Z≥0(0, 1)⊕ Z≥0(0,−1). Because U{0} = hom(S_{0}, T). Define a homomor-phism f : S_{0} → T by f(1, 0) = x1, f (−1, 0) = x2, f (0, 1) = x₃, and

f (0,−1) = x4. Then x1 + x2 = 0 and x3 + x4 = 0, and so x1, x2, x3, and x4 don’t equal −∞. So U{0} = hom(S_{0}, T) � (T^∗)². Hence (T^∗)² is the 2-dimensional algebraic torus over T.

Example 4.2.3. Let σ = pos{(0, −1)} in N^R � R², then the dual cone σ^∨ = pos{(1, 0), (−1, 0), (0, −1)}, and the corresponding semigroup S^σ = Z_≥0(1, 0) ⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1). Let f be in hom(S^σ, T). Suppose that f (1, 0) = x1, f (−1, 0) = x2, and f (0,−1) = x3, then 0 = f (0, 0) = f (0, 1) + f (0,−1) = f(0, 1) ⊗ f(0, −1) = x¹ ⊗ x² = x1 + x2. Hence the corresponding affine toric variety is Uσ = hom(Sσ, T) = R× R.

Proposition 4.2.4. If τ ⊂ σ is a face of a cone σ, then we obtain the embedding hom(Sτ, T) = Uτ �→ Uσ = hom(Sσ, T).

Proof. Because τ ⊂ σ is a face of a cone σ, so σ^∨ is a subset of τ^∨. Since Sσ = σ^∨∩ M is a subset of S^τ = τ^∨∩ M, we have the embedding S^σ �→ S^τ. Because Uτ = hom(Sτ, T) and Uσ = hom(Sσ, T), so hom(Sτ, T) = Uτ �→ Uσ = hom(Sσ, T). Note that we have the following commutative diagram:

Sσ� � ^h ��

f◦h

��Sτ

f ��T

Definition 4.2.5. Let F be a rational fan. Then the tropical toric variety

X_F(T) is defined as the quotient

Note that we have the following commutative diagram:

U_τ∩σ^⊂h2> Uσ

Definition 4.2.7. Let XF(T) be a tropical toric variety. A point p is said to be regular or smooth in XF(T) if there is a unimodular cone σ in F such that p is in U_σ. A tropical toric variety X_F(T) is called regular or smooth if all points of XF(T) are regular or smooth. A tropical toric variety XF(T) is called singular if XF(T) is not regular.

Definition 4.2.8. Let F be fan in NR. A set ∆ is called a subfan of F if ∆ is a subset of F and is also a fan.

Definition 4.2.9. Let the lattice N = Ze1 ⊕ · · · ⊕ Zen, and let another lattice N^� = Ze1⊕ · · · ⊕ Ze^m. Let F be a fan in NR, and let ∆ be another fan in N^�. Let φ : N^R → N^R� be a linear map such that φ(N ) ⊆ N^� and there is a cone σ^� in ∆ containing φ(σ) for all cone σ in F . Then the map φ : F → ∆ is said a map of fans.

Theorem 4.2.10. Let N � Zⁿ and N^� � Z^m be two different lattice, and let M = hom(N, Z) and M^� = hom(N^�, Z) be the dual lattice. Let F be a fan in NR and let ∆ be another fan in NR^� . Let φ : F → ∆ be a map of fans.

Then φ extends to a continuous map φ : XF(T)→ X∆(T).

Proof. Since φ : F → ∆ is a map of fans, there is a cone σ^� in ∆ such that φ(σ)⊂ σ^�. Let u be in Sσ^�. Then u is in hom(N^�, Z) and is in σ^�∨, and thus we have the map u : N^� → Z via u(n^�) =< u, n^� >. Since φ : F → ∆ is a

map of fans, φ(N ) ⊂ N^{� u ��}Z . Let µ : N → Z via µ(n) =< u, φ(n) >, then µ is in hom(N, Z). Since φ(σ) is a subset of σ^�, φ(n) is in σ^� for all n in σ. Since u is in σ^�∨, < u, φ(n) >≥ 0, and this implies µ is in σ^∨. So µ is in σ^∨∩ hom(N, Z) = Sσ. Therefore, we obtain a map S_σ� → Sσ, and thus have a map Uσ → Uσ^�. And we have the following commutative diagram:

Uσ1 < ^⊃Uσ1∩σ2 ⊂ > Uσ2

Uσ₁^�

∨ < ^⊃Uσ^�₁∩σ^�2

∨

⊂ > Uσ₂^�

∨

Hence XF(T)→ X∆(T).

Definition 4.2.11. For all r in Z, we define the tropical Hirzebruch surface Hr to be

THr:= {([x0 : x₁], [y₀ : y₁ : y₂])∈ TP¹ × TP² | rx0+ y₀ = rx₁+ y₁}.

For more information on tropical Hirzebruch surfaces, see [3] § 2.

Figure 4.1: the fan ∆

Figure 4.2: the polytope P

4.3 Smooth two-dimensional tropical toric Fano varieties

In this section, we know that there are only five smooth Fano polytopes in R² up to the action of GL(2, Z), so I will calculate these cases of smooth two-dimensional tropical toric Fano varieties.

Example 4.3.1. Given the lattice N = Z(1, 0)⊕ Z(0, 1) � Z², then N^R = N ⊗ R � R², the dual lattice M � Z² and MR= M ⊗ R.

Let the fan ∆ in N^R. Suppose that the fan ∆ (the figure 4.1 ) has

σ1 = pos{(1, 0), (0, 1)}, σ² = pos{(−1, 0), (0, 1)},

σ₃ = pos{(−1, 0), (0, −1)}, σ4 = pos{(1, 0), (0, −1)},

together with

τ1 = σ1∩ σ2 = pos{(0, 1)}, τ2 = σ2∩ σ3 = pos{(−1, 0)},

τ3 = σ3∩ σ⁴ = pos{(0, −1)}, τ⁴ = σ4 ∩ σ¹ = pos{(1, 0)},

and the origin. Then the dual cones

σ₁^∨ = pos{(1, 0), (0, 1)}, σ2^∨ = pos{(−1, 0), (0, 1)},

σ^∨₃ = pos{(−1, 0), (0, −1)}, σ4^∨ = pos{(1, 0), (0, −1)}.

Moreover, the corresponding semigroups

Sσ1 = σ₁^∨∩ M = Z≥0(1, 0)⊕ Z≥0(0, 1),

Sσ2 = σ₂^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(0, 1),

Sσ3 = σ₃^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(0,−1),

S_σ4 = σ₄^∨∩ M = Z≥0(1, 0)⊕ Z≥0(0,−1),

together with

Sτ1 = Sσ1 + Sσ2 = Z_≥0(1, 0)⊕ Z≥0(−1, 0) ⊕ Z≥0(0, 1),

Sτ2 = Sσ2+ Sσ3 = Z_≥0(0, 1)⊕ Z≥0(0,−1) ⊕ Z≥0(−1, 0),

Sτ3 = Sσ3+ Sσ4 = Z_≥0(1, 0)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1),

Sτ4 = Sσ4 + Sσ1 = Z_≥0(0, 1)⊕ Z≥0(0,−1) ⊕ Z≥0(1, 0),

S_{0}= Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1).

Let fi be in Uσi = hom(Sσi, T) for all i = 1, 2, 3, 4, then we have some maps

f1 : Sσ1 → T via f¹(1, 0) = x and f1(0, 1) = y,

f2 : Sσ2 → T via f²(−1, 0) = −x and f²(0, 1) = y,

f3 : Sσ3 → T via f3(−1, 0) = −x and f3(0,−1) = −y,

f4 : Sσ4 → T via f4(1, 0) = x and f4(0,−1) = −y.

Therefore, the affine toric varieties

Uσ1 = hom(Sσ1, T) = T², Uσ2 = hom(Sσ2, T) = T²,

Uσ3 = hom(Sσ3, T) = T², Uσ4 = hom(Sσ4, T) = T²,

together with

Uτ1 = hom(Sτ1, T) = R× T, Uτ2 = hom(Sτ2, T) = R× T,

Uτ3 = hom(Sτ3, T) = R× T, Uτ4 = hom(Sτ4, T) = R× T,

U_{0}= hom(S_{0}, T) = R².

The gluing of the affine toric varieties Uσ1 and Uσ2 along their common subset Uτ1 gives TP¹ × T with coordinates ((x⁰ : x1), y) where x = x1− x⁰. The gluing of the affine toric varieties U_σ2 and U_σ3 along their common subset Uτ2 gives T× TP¹ with coordinates (−x, (y0 : y1)) where y = y1 − y0. The gluing of the affine toric varieties Uσ3 and Uσ4 along their common subset U_τ3 gives TP¹ × T with coordinates ((x0 : x₁),−y) where x = x1− x0. The gluing of the affine toric varieties Uσ4 and Uσ1 along their common subset Uτ4 gives T× TP¹ with coordinates (x, (y0 : y1)) where y = y1 − y⁰. The

following commutative diagram:

Hence the gluing of these two gives the tropical toric variety

X_∆(T) = (� basis of N , XP is a smooth Fano polytope (by Theorem 3.4.2).

Example 4.3.2. Given the lattice N � Z², then N^R = N ⊗ R � R², the dual lattice M � Z² and MR = M⊗ R.

Let the fan ∆ in N^R. Suppose that the fan ∆ (the figure 4.3) has

σ1 = pos{(1, 0), (0, 1)}, σ² = pos{(−1, −1), (0, 1)}, σ³ = pos{(1, 0), (−1, −1)},

Figure 4.3: the fan ∆

Figure 4.4: the polytope P

together with

τ1 = σ1∩ σ² = pos{(0, 1)}, τ² = σ2∩ σ³ = pos{(−1, −1)},

τ3 = σ3∩ σ¹ = pos{(1, 0)}, and the origin.

Then the dual cones

σ₁^∨ = pos{(1, 0), (0, 1)}, σ2^∨ = pos{(−1, 0), (−1, 1)}, σ^∨3 = pos{(1, −1), (0, −1)}.

Moreover, the corresponding semigroups

Sσ1 = σ₁^∨∩ M = Z≥0(1, 0)⊕ Z≥0(0, 1),

Sσ2 = σ₂^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(−1, 1),

S_σ3 = σ₃^∨∩ M = Z≥0(1,−1) ⊕ Z≥0(0,−1),

together with

S_τ1 = S_σ1 + S_σ2 = Z_≥0(1, 0)⊕ Z≥0(−1, 0) ⊕ Z≥0(0, 1),

Sτ2 = Sσ2 + Sσ3 = Z_≥0(1,−1) ⊕ Z≥0(−1, 1) ⊕ Z≥0(−1, −1),

S_τ3 = S_σ3 + S_σ1 = Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(0,−1),

S_{0}= Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1).

Let fi be in Uσi = hom(Sσi, T) for all i = 1, 2, 3, then we have some maps

f1 : Sσ1 → T via f¹(1, 0) = x and f1(0, 1) = y,

f2 : Sσ2 → T via f²(−1, 0) = −x and f²(−1, 1) = −x + y,

f3 : Sσ3 → T via f³(1,−1) = x − y and f³(0,−1) = −y,

Therefore, the affine toric variety

Uσ1 = hom(Sσ1, T) = T², Uσ2 = hom(Sσ2, T) = T²,

Uσ3 = hom(Sσ3, T) = T²,

together with

Uτ1 = hom(Sτ1, T) = R× T, Uτ2 = hom(Sτ2, T) = R× T,

Uτ3 = hom(Sτ3, T) = R× T, U{0} = hom(S_{0}, T) = R².

The gluing of the affine toric varieties Uσ1 and Uσ2 along their common subset U_τ1 gives TP² with coordinates (x₀ : x₁ : x₂) where x = x₁ − x0

Hence the gluing of these two gives the tropical toric variety

X_∆(T) = (�

σ∈∆

Uσ)/∼= TP².

The polytopes P = conv{0, (1, 0), (0, 1), (−1, −1)} (the figure 4.4) in N^R.Since conv{(1, 0), (0, 1)}, conv{(0, 1), (−1, −1)}, conv{(−1, −1), (1, 0)}

are the facets of P , and they are the convex hull of a basis of N , XP is a smooth Fano polytope (by Theorem 3.4.2).

Example 4.3.3. Given the lattice N � Z², then N^R = N ⊗ R � R², the dual lattice M � Z² and MR = M⊗ R.

Let the fan ∆ in N^R. Suppose that the fan ∆ (the figure 4.5) has

σ1 = pos{(1, 0), (1, 1)}, σ² = pos{(1, 1), (0, 1)},

σ₃ = pos{(0, 1), (−1, −1)}, σ4 = pos{(−1, −1), (1, 0)},

together with

τ1 = σ1∩ σ² = pos{(1, 1)}, τ² = σ2 ∩ σ³ = pos{(0, 1)},

τ₃ = σ₃∩ σ4 = pos{(−1, −1)}, τ4 = σ₄ ∩ σ1 = pos{(1, 0)},

Figure 4.5: the fan ∆

Figure 4.6: the polytope P

and the origin. Then the dual cones

σ^∨₁ = pos{(−1, −1), (0, 1)}, σ2^∨ = pos{(1, 0), (−1, 1)},

σ^∨₃ = pos{(−1, 0), (−1, 1)}, σ4^∨ = pos{(0, −1), (1, −1)}.

Moreover, the corresponding semigroups

Sσ1 = σ₁^∨∩ M = Z≥0(0, 1)⊕ Z≥0(1,−1),

Sσ2 = σ₂^∨∩ M = Z≥0(1, 0)⊕ Z≥0(−1, 1),

Sσ3 = σ₃^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(−1, 1),

Sσ4 = σ₄^∨∩ M = Z≥0(0,−1) ⊕ Z≥0(1,−1),

together with

Sτ1 = Sσ1+ Sσ2 = Z_≥0(1, 1)⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ2 = Sσ2 + Sσ3 = Z_≥0(0, 1)⊕ Z≥0(1, 0)⊕ Z≥0(−1, 0),

Sτ3 = Sσ3 + Sσ4 = Z_≥0(−1, −1) ⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ4 = Sσ4 + Sσ1 = Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(0,−1),

S_{0}= Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1).

Let fi be in Uσi = hom(Sσi, T) for all i = 1, 2, 3, 4, then we have some maps

f1 : Sσ1 → T via f¹(0, 1) = y and f1(1,−1) = x − y,

f2 : Sσ2 → T via f²(1, 0) = x and f2(−1, 1) = −x + y,

f3 : Sσ3 → T via f3(−1, 0) = −x and f3(−1, 1) = −x + y,

f4 : Sσ4 → T via f4(0,−1) = −y and f4(1,−1) = x − y.

Therefore, the affine toric variety

Uσ1 = hom(Sσ1, T) = T², Uσ2 = hom(Sσ2, T) = T²,

Uσ3 = hom(Sσ3, T) = T², Uσ4 = hom(Sσ4, T) = T²,

together with

Uτ1 = hom(Sτ1, T) = R× T, Uτ2 = hom(Sτ2, T) = R× T,

Uτ3 = hom(Sτ3, T) = R× T, Uτ4 = hom(Sτ4, T) = R× T,

U_{0}= hom(S_{0}, T) = R².

The gluing of the affine toric varieties Uσ2 and Uσ3 along their common subset Uτ2 gives TP¹ × T with coordinates ((x⁰ : x1),−x + y) where x = x₀ − x1. The gluing of the affine toric varieties U_σ4 and U_σ1 along their common subset Uτ4 gives TP¹ × T with coordinates ((y0 : y1), x− y) where y = y0− y¹.

The two copies of TP¹×T are glued along their second coordinates gives TP¹× TP² with coordinates ([z0 : z1], [x0− x¹ : y0− y¹ : 0]) = ([z0 : z1], [x0− x1 : y0− y1 : 0]) where x− y = z0− z1. Since z0− z1 = (x0− x1)− (y0− y1), (x0− x1)− z0 = (y0− y1)− z1. Hence the toric variety

X_∆(T) = (�

σ∈∆U_σ)�

∼

={([z⁰ : z1], [x0− x¹ : y0− y¹ : 0]) ∈ TP¹× TP² | (x⁰− x¹)− z⁰ =

(y0− y¹)− z¹},

={([z0 : z1], [x0+ y1 : y0+ x1 : x1+ y1])∈ TP¹× TP² | (x0− x1)− z0 = (y0− y1)− z1},

= TH₋₁

where TH₋₁ is a tropical Hirzebruch surf ace.

The following commutative diagram: basis of N , XP is a smooth Fano polytope (by Theorem 3.4.2).

Example 4.3.4. Given the lattice N � Z², then N^R = N ⊗ R � R², the dual lattice M � Z² and MR = M⊗ R.

Let the fan ∆ in N^R. Suppose that the fan ∆ (the figure 4.7) has

σ1 = pos{(1, 0), (1, 1)}, σ2 = pos{(1, 1), (0, 1)},

σ3 = pos{(0, 1), (−1, −1)}, σ⁴ = pos{(−1, −1), (0, −1)},

σ5 = pos{(0, −1), (1, 0)},

together with

τ1 = σ1∩ σ2 = pos{(1, 1)}, τ2 = σ2 ∩ σ3 = pos{(0, 1)},

τ3 = σ3∩ σ⁴ = pos{(−1, −1)}, τ⁴ = σ4 ∩ σ⁵ = pos{(0, −1)},

τ5 = σ5∩ σ1 = pos{(1, 0)},

and the origin. Then the dual cones

σ^∨₁ = pos{(1, −1), (0, 1)}, σ2^∨ = pos{(1, 0), (−1, 1)},

σ^∨₃ = pos{(−1, 0), (−1, 1)}, σ4^∨ = pos{(−1, 0), (1, −1)},

σ^∨₅ = pos{(0, −1), (1, 0)}.

Moreover, the corresponding semigroups

Sσ1 = σ₁^∨∩ M = Z≥0(0, 1)⊕ Z≥0(1,−1),

Sσ2 = σ₂^∨∩ M = Z≥0(1, 0)⊕ Z≥0(−1, 1),

Sσ3 = σ₃^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(−1, 1),

Sσ4 = σ₄^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(1,−1),

S_σ5 = σ₅^∨∩ M = Z≥0(0,−1) ⊕ Z≥0(1, 0),

together with

S_τ1 = S_σ1+ S_σ2 = Z_≥0(1, 1)⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ2 = Sσ2 + Sσ3 = Z_≥0(0, 1)⊕ Z≥0(1, 0)⊕ Z≥0(−1, 0),

Sτ3 = Sσ3 + Sσ4 = Z_≥0(−1, −1) ⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ4 = Sσ4+ Sσ5 = Z_≥0(0,−1) ⊕ Z≥0(1, 0)⊕ Z≥0(−1, 0),

Sτ5 = Sσ5 + Sσ1 = Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(0,−1),

S_{0}= Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1).

Therefore, the affine toric variety

Uσ1 = hom(Sσ1, T) = T², Uσ2 = hom(Sσ2, T) = T²,

Uσ3 = hom(Sσ3, T) = T², Uσ4 = hom(Sσ4, T) = T²,

Uσ5 = hom(Sσ5, T) = T²,

together with

Uτ1 = hom(Sτ1, T) = R× T, Uτ2 = hom(Sτ2, T) = R× T,

Uτ3 = hom(Sτ3, T) = R× T, U^τ4 = hom(Sτ4, T) = R× T,

Uτ5 = hom(Sτ5, T) = R× T, U{0} = hom(S_{0}, T) = R².

The gluing of the affine toric varieties Uσ2 and Uσ3 along their common subset Uτ2 gives TP¹×T with coordinates ([x⁰ : x1],−x+y) where x = x⁰−x¹. The gluing of the affine toric varieties Uσ3 and Uσ4 along their common subset Uτ3 gives TP¹× T with coordinates (−x, [z⁰ : z1]) where −x + y = z¹ − z⁰. The two copies of TP¹×T are glued along their coordinates gives TP¹×TP¹ with coordinates ([x0 : x1], [z0 : z1]). Since we have some embedding h44 : Uτ4 �→ U^σ4 via h44(x0 − x¹, x1− x⁰, x1− x⁰+ z0− z¹) = (0, x1− x⁰, z0− z¹)

and h45 : Uτ4 �→ Uσ5 via h45(x0 − x1, x1 − x0, x1 − x0 + z0 − z1) = (x0 −

Figure 4.7: the fan ∆

Figure 4.8: the polytope P

conv{(−1, −1), (0, −1)}, conv{(0, −1), (1, 0)} are the facets of P , and they are the convex hull of a basis of N , XP is a smooth Fano polytope (by Theorem 3.4.2).

Example 4.3.5. Given the lattice N � Z², then N^R = N ⊗ R � R², the dual lattice M � Z² and M^R = M⊗ R.

Let the fan ∆ in N^R. Suppose that the fan ∆ (the figure 4.9) has

σ1 = pos{(1, 0), (1, 1)}, σ² = pos{(1, 1), (0, 1)},

σ3 = pos{(0, 1), (−1, 0)}, σ⁴ = pos{(−1, 0), (−1, −1)},

σ5 = pos{(−1, −1), (0, −1)}, σ⁶ = pos{(0, −1), (1, 0)},

Figure 4.9: the fan ∆

Figure 4.10: the polytope P

together with

τ1 = σ1∩ σ² = pos{(1, 1)}, τ² = σ2 ∩ σ³ = pos{(0, 1)},

τ3 = σ3∩ σ⁴ = pos{(−1, 0)}, τ⁴ = σ4∩ σ⁵ = pos{(−1, −1)},

τ5 = σ5∩ σ6 = pos{(0, −1)}, τ6 = σ6 ∩ σ1 = pos{(1, 0)},

and the origin. Then the dual cones

σ^∨₁ = pos{(−1, −1), (0, 1)}, σ2^∨ = pos{(1, 0), (−1, 1)},

σ^∨₃ = pos{(−1, 0), (0, 1)}, σ4^∨ = pos{(−1, 1), (0, −1)},

σ^∨₅ = pos{(−1, 0), (1, −1)}, σ6^∨ = pos{(0, −1), (1, 0)}.

Moreover, the corresponding semigroups

Sσ1 = σ₁^∨∩ M = Z≥0(0, 1)⊕ Z≥0(1,−1),

Sσ2 = σ₂^∨∩ M = Z≥0(1, 0)⊕ Z≥0(−1, 1),

Sσ3 = σ₃^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(0, 1),

Sσ4 = σ₄^∨∩ M = Z≥0(−1, 1) ⊕ Z≥0(0,−1),

S_σ5 = σ₅^∨∩ M = Z≥0(−1, 0) ⊕ Z≥0(1,−1),

Sσ6 = σ₆^∨∩ M = Z≥0(0,−1) ⊕ Z≥0(1, 0),

together with

Sτ1 = Sσ1+ Sσ2 = Z_≥0(1, 1)⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ2 = Sσ2 + Sσ3 = Z_≥0(0, 1)⊕ Z≥0(1, 0)⊕ Z≥0(−1, 0),

Sτ3 = Sσ3+ Sσ4 = Z_≥0(−1, 0) ⊕ Z≥0(0, 1)⊕ Z≥0(0,−1),

Sτ4 = Sσ4 + Sσ5 = Z_≥0(−1, −1) ⊕ Z≥0(1,−1) ⊕ Z≥0(−1, 1),

Sτ5 = Sσ5+ Sσ6 = Z_≥0(0,−1) ⊕ Z≥0(1, 0)⊕ Z≥0(−1, 0),

Sτ6 = Sσ6 + Sσ1 = Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(0,−1),

S_{0}= Z_≥0(1, 0)⊕ Z≥0(0, 1)⊕ Z≥0(−1, 0) ⊕ Z≥0(0,−1).

Therefore, the affine toric variety

U_σ1 = hom(S_σ1, T) = T², U_σ2 = hom(S_σ2, T) = T²,

Uσ3 = hom(Sσ3, T) = T², Uσ4 = hom(Sσ4, T) = T²,

U_σ5 = hom(S_σ5, T) = T², U_σ6 = hom(S_σ6, T) = T²,

together with

U_τ1 = hom(S_τ1, T) = R× T, Uτ2 = hom(S_τ2, T) = R× T,

Uτ3 = hom(Sτ3, T) = R× T, Uτ4 = hom(Sτ4, T) = R× T,

Uτ5 = hom(Sτ5, T) = R× T, U^τ6 = hom(Sτ6, T) = R× T,

U_{0}= hom(S_{0}, T) = R².

Let fi be in Uσ_i = hom(Sσ_i, T) for all i = 1, 2, 3, 4, 5, 6, then we have

some maps

f1 : Sσ1 → T via f1(0, 1) = y and f1(1,−1) = x − y,

f2 : Sσ2 → T via f²(1, 0) = x and f2(−1, 1) = −x + y,

f3 : Sσ3 → T via f3(−1, 0) = −x and f3(0, 1) = y,

f4 : Sσ4 → T via f4(0,−1) = −y and f4(1,−1) = −x + y,

f₅ : S_σ4 → T via f4(−1, 0) = −x and f4(1,−1) = x − y,

f5 : Sσ4 → T via f⁴(0,−1) = −y and f⁴(1, 0) = x.

Since we have some embedding h₁₁ : U_τ1 �→ Uσ1 via h₁₁(x + y, x − y,−x + y) = (0, x − y, y) and h12 : Uτ1 �→ Uσ2 via h12(x + y, x− y, −x + y) = (x,−x + y, 0), we have tropical isomorphism h¹² ◦ h⁻¹11 : Uσ1 → U^σ2 via h₁₂ ◦ h⁻¹11(0, x − y, y) = (x, −x + y, 0). Similarly, h23 ◦ h⁻¹22 : U_σ2 → Uσ3

via h23 ◦ h⁻¹22(x,−x + y, 0) = (−x, 0, y), and h34 ◦ h⁻¹33 : Uσ3 → Uσ4 via h34◦h⁻¹33(−x, 0, y) = (0, −x+y, −y), h⁴⁵◦h⁻¹44 : Uσ4 → U^σ5 via h45◦h⁻¹44(0,−x+

y,−y) = (−x, x − y, 0), h56◦ h⁻¹55 : U_σ5 → Uσ6 via h₅₆◦ h⁻¹55(−x, x − y, 0) = (x, 0,−y), h61◦h⁻¹66 : Uσ6 → Uσ1 via h61◦h⁻¹66(x, 0,−y) = (0, x−y, y). Consider the product TP¹×TP¹×TP¹×TP²with homogeneous coordinates [x0 : x1],

[z0 : z1],[y0 : y1] on respective TP¹ and homogeneous coordinate [a : b : c] on TP². We set x = x0 − x¹, x− y = z⁰ − z¹, y = y0 − y¹. Hence the toric variety

X_∆(T) = (

�

σ∈∆Uσ)�

∼

={([x0 : x₁], [z₀ : z₁], [y₀, y₁], [a : b : c])∈ TP¹× TP¹× TP¹

× TP² | b + x¹ = c + x0, a + z1 = b + z0, c + y1 = a + y0},

that is, X_∆(T) is the blow up of TP² at the three points [0 : −∞ : −∞], [−∞ : 0 : −∞], and [−∞ : −∞ : 0].

The polytopes P = conv{0, (1, 0), (1, 1), (0, 1), (0, −1), (−1, −1), (0, −1)}

(the figure 4.10 ) in N^R. Since conv{(1, 0), (1, 1)}, conv{(0, 1), (1, 1)}, conv{(0, 1), (−1, 0)}, conv{(−1, 0), (−1, −1)}, conv{(−1, −1), (0, −1)}, conv{(0, −1), (1, 0)} are the facets of P , and they are the convex hull of a basis of N , XP is a smooth Fano polytope (by Theorem 3.4.2).

Bibliography

[1] Danko Adrovic and Jan Verschelde. Tropical algebraic geometry in maple, a preprocessing algorithm for finding common factors to mul-tivariate polynomials with approximate coefficients. September 2008.

[2] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas.

Computing tropical varieties. J. Symbolic Comput., 42(1-2):54–73, 2007.

[3] E. Brugall´e. Deformation of tropical hirzebruch surfaces. http:

//www.math.jussieu.fr/~brugalle/articles/TropDegHirzebruch/

TropDeg.pdf.

[4] W. Bruns and J. Gubeladze. Polytopes, Rings, and K-Theory. Number 978 0-76352 in Springer Monographs in Mathematics. Springer, 2009.

[5] D.A. Cox, J.B. Little, and H.K. Schenck. Toric Varieties. Graduate Studies in Mathematics. American Mathematical Society, 2011.

[6] David Eisenbud and Joe Harris. The Geometry of Schemes. Springer, November 2001.

[7] G¨unter Ewald. Combinatorial Convexity and Algebraic Geometry (Grad-uate Texts in Mathematics). Springer, October 1996.

[8] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton, NJ, 1993.

[9] Andreas Gathmann. Algebraic geometry, 2002.

[10] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch.

Math.-Verein., 108(1):3–32, 2006.

[11] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Modern Birkh¨auser Classics. Birkh¨auser, 2008.

[12] Branko Gr¨unbaum. Convex Polytopes : Second Edition Prepared by Volker Kaibel, Victor Klee, and G¨unter Ziegler (Graduate Texts in Math-ematics). Springer, May 2003.

[13] Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin. Tropical algebraic geometry, volume 35 of Oberwolfach Seminars. Birkh¨auser Verlag, Basel, second edition, 2009.

[14] A Kasprzyk. Toric Fano Varieties and Convex Polytopes. PhD thesis, 2006.

[15] Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geome-try. October 9, 2009.

[16] G.G. Magaril-Ilyaev and V.M. Tikhomirov. Convex Analysis: Theory and Applications. Translations of Mathematical Monographs. American Mathematical Society, 2003.

[17] H. Meyer. Intersection Theory on Compact Tropical Toric Varieties.

Sudwestdeutscher Verlag F R Hochschulschriften AG, 2011.

[18] G. Mikhalkin. Amoebas of algebraic varieties and tropical geometry.

ArXiv Mathematics e-prints, February 2004.

[19] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R². J.

Amer. Math. Soc., 18(2):313–377 (electronic), 2005.

[20] Grigory Mikhalkin. Tropical geometry and its applications. In Interna-tional Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math.

Soc., Z¨urich, 2006.

[21] Grigory Mikhalkin and Ilia Zharkov. Tropical curves, their jacobians and theta functions, November 2007.

[22] T. Oda. Convex bodies and algebraic geometry: an introduction to the theory of toric varieties : with 42 figures. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1988.

[23] J¨urgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer. Math.

Soc., Providence, RI, 2005.

[24] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.

[25] K.E. Smith. An Invitation to Algebraic Geometry. Universitext (1979).

Springer, 2000.

[26] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009.

[27] Thorsten Theobald and Thorsten Theobald. Computing amoebas. Ex-perimental Math, 11:513–526, 2002.

[28] Gunter M. Ziegler. Lectures on Polytopes (Graduate Texts in Mathe-matics). Springer, November 1994.

在文檔中 二維平滑熱帶環面法諾曲體之研究 - 政大學術集成 (頁 56-106)