A Reider-type Result for Smooth Projective Toric Surfaces

TL;DR

This paper establishes numerical criteria for when the adjoint line bundle on a smooth projective toric surface is nef or ample, based on the associated lattice polytope.

Contribution

It provides new necessary and sufficient conditions for the positivity of the adjoint series on toric surfaces using polytope analysis.

Findings

Criteria for nefness of |K_X+L|

Criteria for ampleness of |K_X+L|

Polytope-based numerical conditions

Abstract

Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be either nef or (very) ample.

Equations

$$D\cdot L=1$$

$$D\cdot L=2$$

$$D\cdot L=3$$

$$D\cdot L=1$$

$$D\cdot L=2$$

$$A\cdot C=|E\cap M|-1.$$

$$K_X\cdot D_i=b_i-2,$$

$$(L+K_X)\cdot D_i=L\cdot D_i-D_i^2-2.$$

$$0\sim div({\chi }^{e_1})$$

$$0\sim div({\chi }^{e_2})$$

$$Pic({\mathcal{F}}_r)\simeq \left\{a{D}_3+b{D}_4|a,b\in \mathbb{Z}\right\}.$$

$$m_1=(-a,0),m_2=(-a,b),m_3=(rb,b),m_4=(0,0).$$

$$(D\cdot E{)}^2\ge D^2{E}^2.$$

$$D\cdot H=(D^2)E\cdot D-(D\cdot E)D^2=0.$$

$$0\ge$$

$$=$$

$$Vol(P)=|\partial P\cap M|+2|P^0\cap M|-2\ge 9.$$

$$Vol(P)=|\partial P\cap M|+2|P^0\cap M|-2\ge 10.$$

$$L^2\ge L\cdot D_{{\rho }_i}+4.$$

$$vol(P)=\frac{L^2}{2}=\frac{|\partial P\cap M|}{2}+|P^0\cap M|-1,$$

$$|\partial P\cap M|=\sum _{j=1}^{n}L\cdot D_{{\rho }_j}.$$

$$L^2=\sum _{j=1}^{n}L\cdot D_{{\rho }_j}+2|P^0\cap M|-2\ge L\cdot D_{{\rho }_i}+(n-1)+2|P^0\cap M|-2.$$

$$L^2\ge L\cdot D_{{\rho }_i}+4.$$

$$(L+K_X)\cdot D_{\rho }=L\cdot D_{\rho }-D_{\rho }^2-2<0.$$

$$(L+K_X)\cdot D_{\rho }=L\cdot D_{\rho }-D_{\rho }^2-2\le 0.$$

$$Pic({\mathcal{F}}_r)\simeq \left\{a{D}_3+b{D}_4|a,b\in \mathbb{Z}\right\}.$$

$$K_X=-(D_1+D_2+D_3+D_4)\sim -(2-a)D_3-2D_4.$$

$$L\cdot D=1\text{ and }{D}^2=0.$$

$$L\cdot D=1\text{ and }{D}^2=-1.$$

$$L\cdot D=1\text{ and }{D}^2=0.$$

$$L\cdot D=2\text{ and }{D}^2=0.$$

Full text

1. Introduction

The problem of determining whether a line bundle is nef or (very) ample is an important question in algebraic geometry. The Nakai-Moishezon criterion [12, 10] states that a Cartier divisor $D$ on a proper scheme $X$ over an algebraically closed field is ample if and only if $D^{\dim(Y)}\cdot Y>0$ for every closed integral subscheme $Y$ of $X$. For toric varieties, a special form of the criterion holds: if $D\cdot C>0$ for every torus-invariant curve $C\subset X$ then $D$ is ample. Furthermore, if $D\cdot C\geq 0$ for every torus-invariant curve $C\subset X$ then $D$ is globally generated [7, 9, 11]. However, the question is more complicated when we consider the adjoint bundle $D+K_{X}$. Namely, are there numerical conditions for $D\cdot C$ so that $D+K_{X}$ is globally generated or ample? Fujita conjectured the following:

Conjecture 1.1** ([3]).**

Let $X$ be an $n$-dimensional projective algebraic variety, smooth or with mild singularities, and $D$ an ample divisor on $X$. Then

1. (1)

For $t\geq n+1$, $tD+K_{X}$ is basepoint free. 2. (2)

For $t\geq n+2$, $tD+K_{X}$ is very ample.

The conjecture is true for toric varieties [4, 13]. For smooth surfaces, Fujita’s conjecture follows from Reider’s theorem [15].

In this article, we will present a combinatorial proof for a Reider-type result for smooth projective toric surfaces.

Proposition 1.2**.**

Let $X$ be a smooth projective toric surface not isomorphic to $\mathbb{P}^{2}$, and let $L$ be an ample line bundle on $X$.

1. (1)

The adjoint series $|K_{X}+L|$ is not base point free if and only if there exists an effective torus-invariant divisor $D\subset X$ such that

[TABLE] 2. (2)

The adjoint series $|K_{X}+L|$ is not ample if and only if there exists an effective torus-invariant divisor $D\subset X$ such that either

[TABLE]

Furthermore, if $L^{2}\geq 10$, then $|K_{X}+L|$ is not ample if and only if there exists an effective torus-invariant divisor $D\subset X$ such that either

[TABLE]

As a convention, in this article, we will follow the notations in [2]. In particular, we will always use $M$ to denote the ambient lattice if there is no confusions.

Acknowledgments

We would like to thank Milena Hering for suggesting the problem and for her invaluable guidance. We also want to thank Ivan Cheltsov for some of the comments.

2. Toric Surfaces Reviewed

Let $A$ be an ample line bundle over a projective toric variety $X$ corresponding to a polytope $P\subset M_{\mathbb{R}}$. Then we have a combinatorial interpretation of the intersection number $A\cdot C$ where $C\subset X$ is any torus-invariant curve as follows.

Lemma 2.1** ([7, (1.4) and Page 457]).**

Let $A$ be an ample line bundle on a projective toric variety $X$ corresponding to a polytope $P$. For a torus invariant curve $C$, let $E$ be the corresponding edge on $P$. Then $A\cdot C$ is equal to the lattice length of $E$, i.e.,

[TABLE]

For our purpose, we will need to use the classification of smooth projective toric surfaces: every smooth complete toric surfaces is a finite blowup of either $\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$, or the Hirzebruch surface $\mathscr{F}_{a}$, where $a\geq 2$ ( [2, Theorem 10.4.3]). Another important fact that we will use is that every ample line bundle on a smooth projective toric surface is also very ample.

Lemma 2.2** ([2, Theorem 6.1.15]).**

A line bundle on a smooth complete toric variety is ample if and only if it is very ample.

Smooth toric surfaces are interesting objects to work with; partially because of their computability. For example, we have the following lemma.

Lemma 2.3** ([2, Proposition 10.4.11]).**

Let $u_{0},\ldots,u_{r}$ be ray generators of a smooth complete fan $\Sigma$ in $N_{\mathbb{R}}\cong\mathbb{R}^{2}$. Let $X=X_{\Sigma}$ be the smooth projective toric surface from $\Sigma$ and $D_{i}=V(u_{i})$ for $0\leq i\leq r$. Let $K_{X}$ be the canonical divisor $K_{X}=-\sum_{i=0}^{r}D_{i}$. Then

[TABLE]

where the $b_{1},\ldots,b_{r-1}$ are integers such that $u_{i-1}+u_{i+1}=b_{i}u_{i}$ for all $0\leq i\leq r$, where $u_{-1}=u_{r}$ and $u_{r+1}=u_{0}$.

The following corollary follows directly from [2, Lemma 10.4.1] and Lemma 2.3.

Corollary 2.4**.**

Let $u_{0},\ldots,u_{r}$ be ray generators of a smooth complete fan $\Sigma$ in $N_{\mathbb{R}}\cong\mathbb{R}^{2}$. Let $X=X_{\Sigma}$ be the smooth projective toric surface from $\Sigma$ and $D_{i}=V(u_{i})$ for $0\leq i\leq r$. Let $K_{X}$ be the canonical divisor $K_{X}=-\sum_{i=0}^{r}D_{i}$. Then for $0\leq i\leq r$,

[TABLE]

We also know that the blowup of a toric variety corresponds to a subdivision of fan. Thus the number of generating rays of the fan corresponding to a toric surface increases after a blowup ([2, Proposition 3.3.15]).

Example 2.5*.*

Consider the Hirzebruch surface $\mathscr{F}_{r}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(r))$, $r\geq 1$, whose fan $\Sigma$ given by the following figure

The ray generators of $\Sigma$ are $v_{1}=(1,0)$, $v_{2}=(0,1)$, $v_{3}=(-1,r)$, and $v_{4}=(0,-1)$. Let the associated divisors be $D_{1}$, $D_{2}$, $D_{3}$, and $D_{4}$, respectively. By [2, Proposition 4.1.2],

[TABLE]

Thus $D_{3}\sim D_{1}$, $D_{4}\sim D_{2}+aD_{3}$, and

[TABLE]

The maximal cones of $\Sigma$ are $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ and $\sigma_{4}$ as in Figure 1. Let $D=aD_{3}+bD_{4}$. We compute the $m_{\sigma_{i}}$ to be

[TABLE]

Then by [2, Lemma 6.1.13], $D$ is very ample if and only if $a,b>0$. The nef cone of $\mathscr{F}_{r}$ is given by

By [2, Lemma 10.4.1], we have $D_{1}^{2}=D_{3}^{2}=0$, $D_{2}^{2}=-a$, $D_{4}^{2}=a$, $D_{1}\cdot D_{2}=D_{2}\cdot D_{3}=D_{3}\cdot D_{4}=D_{4}\cdot D_{1}=1$, $D_{1}\cdot D_{3}=D_{2}\cdot D_{4}=0$.

Finally, we will make use of the Hodge’s Index Theorem:

Lemma 2.6** ([5, Theorem V.1.9]).**

Let $D$ be an ample divisor on a smooth projective surface $S$. If $E$ is a divisor such that $D\cdot E=0$, then $E^{2}\leq 0$. The equality occurs if and only if $E$ is numerically equivalent to [math].

Corollary 2.7** ([5, Exercise V.1.9]).**

Let $D$ be an ample divisor on a smooth projective surface $S$ and $E$ an arbitrary divisor. Then

[TABLE]

Proof.

Since $D$ is ample, $D^{2}>0$. Let $H=(D^{2})E-(D\cdot E)D$. We have

[TABLE]

Then by Lemma 2.6, we must have $H^{2}\leq 0$. In other words,

[TABLE]

Since $D^{2}>0$, it follows that $(D\cdot E)^{2}\geq D^{2}E^{2}$. ∎

3. Toric Surfaces and Lattice Polygons

In this section, we review and prove some lemmas on lattice polygons that we will use to the proof of Proposition 1.2.

Lemma 3.1** ([1, Lemma 1]).**

Every lattice polygon with at least 5 edges has at least an interior lattice point.

Lemma 3.2**.**

Let $v_{1},\ldots,v_{5}$ be lattice points such that no three points are collinear. Then there exists a lattice point in $conv(v_{1},\ldots,v_{5})\backslash\{v_{1},\ldots,v_{5}\}$.

Proof.

Let the coordinates of $v_{i}$ be $(x_{i},y_{i})$ for $i=1,\ldots,5$. By the pigeonhole principle, there must be $i\neq j$ such that $x_{i}\equiv x_{j}\pmod{2}$ and $y_{i}\equiv y_{j}\pmod{2}$. Then the midpoint $m$ of $v_{i}v_{j}$ is a lattice point. Since no three points in $\{v_{1},\ldots,v_{5}\}$ are collinear, it follows that $m\in conv(v_{1},\ldots,v_{5})\backslash\{v_{1},\ldots,v_{5}\}$. ∎

As a consequence, we obtain:

Lemma 3.3**.**

Let $P$ be a lattice polygon that has at least $5$ vertices and assume that one of its edges has lattice length $4$. Then $Vol(P)\geq 9$.

Proof.

It suffices to prove the lemma when $P$ is a lattice pentagon. Let $P=conv(v_{1},\ldots,v_{5})$, where $v_{1},\ldots,v_{5}$ are ordered clockwise in $M$. Without loss of generality suppose that the lattice length of the edge joining $v_{1}$ and $v_{5}$ is $4$; i.e., there are $3$ other lattice points $y_{1}$, $y_{2}$, $y_{3}$ in between $v_{1}$ and $v_{5}$.

Consider the polytope $Q=conv(v_{1},v_{2},v_{3},v_{4},y_{1})$. Then by Lemma 3.1, there must be a lattice point $x$ in the interior of $Q$. Then $x$ lies in at most one of the segments $v_{1}v_{3}$, $y_{1}v_{3}$, $y_{2}v_{3}$, $y_{3}v_{3}$, $v_{5}v_{3}$. If $x$ lies in $v_{1}v_{3}$ or if $x$ does not lie in any mentioned segments, consider the set of $5$ points $\{x,v_{3},v_{4},v_{5},y_{1}\}$. By Lemma 3.2, there must be another lattice point $y$ in $P$ that is not the same as the points listed before. If $y\in\partial P$, then $|\partial P\cap M|\geq 9$ and $|P^{0}\cap M|\geq 1$. By Pick’s theorem [14],

[TABLE]

If $y\in P^{0}$, then $|\partial P\cap M|\geq 8$ and $|P^{0}\cap M|\geq 2$. Again, by Pick’s theorem,

[TABLE]

If $x$ lies in $v_{3}y_{1}$ or $v_{3}y_{2}$ then we get such a point $y$ from $conv(x,v_{3},v_{4},v_{5},y_{3})$. If $x$ lies in $v_{3}y_{3}$ or $v_{3}v_{5}$ then we get $y$ from $conv(v_{1},v_{2},v_{3},x,y_{2})$. The same argument follows and we proved the lemma. ∎

We will also need the following lemmas for the proof of Proposition 1.2.

Lemma 3.4**.**

Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Let $\Sigma$ be the fan of $X$. Suppose that $\Sigma$ has $n\geq 5$ rays $\rho_{1},\ldots,\rho_{n}$. Then for any integer $1\leq i\leq n$,

[TABLE]

Proof.

Let $P$ be the polytope associated to $L$. By Pick’s theorem ([14]) and since $L$ is ample so that $L\cdot D_{\rho_{i}}\geq 1$ for all $i$,

[TABLE]

where $\partial P$ and $P^{0}$ are the sets of all boundary points and interior points of $P$, respectively. By Lemma 2.1,

[TABLE]

Hence, combining (1) and (2) gives

[TABLE]

Since $n\geq 5$, by Lemma 3.1, $|P^{0}\cap M|\geq 1$. Therefore,

[TABLE]

∎

4. A Reider-type Result for Toric Surfaces

We will devote this section to prove Proposition 1.2. First of all, it is true for $X\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Lemma 4.1**.**

Proposition 1.2 holds for $X\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Proof.

Let $\Sigma$ be the fan of $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ as follows.

By [2, Lemma 10.4.1], $D_{\rho}^{2}=0$ for all $\rho\in\Sigma(1)$. Thus, we need to show that there exists $\rho$ such that $L\cdot D_{\rho}=1$ in the first part and $L\cdot D_{\rho}\leq 2$ in the second part.

For any ample bundle $L$ on $X$, if $L+K_{X}$ is not basepoint free, then there exists $\rho\in\Sigma(1)$ such that $(L+K_{X})\cdot D_{\rho}<0$. Then By lemma 2.3,

[TABLE]

This implies $0<L\cdot D_{\rho}<D_{\rho}^{2}+2=2$, so that $L\cdot D_{\rho}=1$.

Now suppose that $L+K_{X}$ is not ample and $(L+K_{X})\cdot D_{\rho}\leq 0$. Then By lemma 2.3,

[TABLE]

This implies $1\leq L\cdot D_{\rho}\leq D_{\rho}^{2}+2=2$. Hence, either $L\cdot D_{\rho}=1$ and $D_{\rho}^{2}=0$ or $L\cdot D_{\rho}=2$ and $D_{\rho}^{2}=0$. The conclusion follows. ∎

Secondly, we show that Proposition 1.2 holds for Hirzebruch surfaces.

Lemma 4.2**.**

Proposition 1.2 holds for $X\cong\mathscr{F}_{a}$, $a\geq 1$.

Proof.

Consider the Hirzebruch surface $X=\mathscr{F}_{r}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(r))$, $r\geq 1$ as in Example 2.5. We have

[TABLE]

The canonical divisor of $X$ is given by

[TABLE]

Recall that $D_{1}^{2}=D_{3}^{2}=0$, $D_{2}^{2}=-a$, $D_{4}^{2}=a$, $D_{1}\cdot D_{2}=D_{2}\cdot D_{3}=D_{3}\cdot D_{4}=D_{4}\cdot D_{1}=1$, and $D_{1}\cdot D_{3}=D_{2}\cdot D_{4}=0$ (cf. [2, Lemma 10.4.1]).

Let $L$ be an ample line bundle over $\mathscr{F}_{r}$. Then $L^{2}>0$. We have two cases as follows.

• •

If $r=1$ then $K_{X}=-D_{3}-2D_{4}$. For $L$ to be ample while $L+K_{X}$ is not nef, $L$ has to be of the form $L\sim cD_{3}+D_{4}$, $c>0$. In this case, take $D=D_{3}$, then

[TABLE]

For $L$ to be ample while $L+K_{X}$ is not ample, $L$ has to be of the form $L\sim D_{3}+cD_{4}$, or $L\sim cD_{3}+D_{4}$, or $L\sim cD_{3}+2D_{4}$, where $c\geq 1$.

1. (1)

If $L\sim D_{3}+cD_{4}$, take $D=D_{2}$, then

[TABLE] 2. (2)

If $L\sim cD_{3}+D_{4}$, take $D=D_{3}$, then

[TABLE] 3. (3)

If $L\sim cD_{3}+2D_{4}$, take $D=D_{3}$, then

[TABLE]

• •

$r\geq 2$: For $L$ to be ample but $K_{X}+L$ is not nef, $L$ has the form

[TABLE]

Take $D=D_{3}$, then $L\cdot D=1\text{ and }D^{2}=0$.

For $L$ to be ample but $K_{X}+L$ is not, $L$ has the form $L\sim cD_{3}+D_{4}$ or $L\sim cD_{3}+2D_{4}$, where $c\geq 1$.

1. (1)

If $L\sim cD_{3}+D_{4}$, take $D=D_{3}$, then

[TABLE] 2. (2)

If $L\sim cD_{3}+2D_{4}$, take $D=D_{3}$, then

[TABLE]

∎

Finally, we will give the proof for the final case of Proposition 1.2, when $X$ is an arbitrary blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ or the Hirzebruch surface.

Proof of Proposition 1.2.

The sufficiency trivially holds by Corollary 2.4. We now prove the necessity.

By the classification of smooth projective toric surfaces, the proofs for the cases of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ (Lemma 4.1) and $\mathscr{F}_{a}$ (Lemma 4.2), it suffices to prove the proposition in the case that the fan $\Sigma$ of $X$ has at least $5$ rays.

We first prove part 1. Suppose that $K_{X}+L$ is not basepoint free. Then there exists $\rho\in\Sigma(1)$ such that $(K_{X}+L)\cdot D_{\rho}<0$. Take $D=D_{\rho}$. By Lemma 2.3,

[TABLE]

This implies $L\cdot D<D^{2}+2$, so since $L$ is ample,

[TABLE]

• •

If $D^{2}\leq-1$, then $L\cdot D\leq 0$, which is a contradiction to the hypothesis that $L$ is ample.

• •

If $D^{2}=0$, either $D\cdot L=0$ or $D\cdot L=1$. But $D\cdot L>0$ since $L$ is ample. Thus $D\cdot L=1$. The proposition holds for this case.

It remains to show that $D^{2}$ cannot be positive. Since the fan of $X$ contains at least 5 rays, by Lemma 3.4,

[TABLE]

In addition, it follows from Corollary 2.7 that

[TABLE]

Combining (5) with (3) and (4) yields

[TABLE]

This implies $L\cdot D\leq 1$. The only possibility is $L\cdot D=1$. Then by (5), $D^{2}=L^{2}=1$, which is impossible since $L^{2}\geq L\cdot D+4=5$. Therefore, it cannot be the case that $D^{2}>0$.

We now prove the second part of the proposition. Suppose that $K_{X}+L$ is not ample, so there exists $\rho\in\Sigma(1)$ such that $(K_{X}+L)\cdot D_{\rho}\leq 0$. Let $D=D_{\rho}$. By Corollary 2.4,

[TABLE]

This implies $L\cdot D\leq D^{2}+2$; hence,

[TABLE]

• •

If $D^{2}=-1$, then $1\leq L\cdot D\leq 1$, so $L\cdot D=1$.

• •

If $D^{2}=0$, either $D\cdot L=1$ or $D\cdot L=2$.

Now we consider the case that $D^{2}\geq 1$. Since the fan of $X$ contains at least 5 rays, by Lemma 3.4,

[TABLE]

By Corollary 2.7,

[TABLE]

Since $D^{2}\geq 1$, then by (7), $L^{2}\geq 5$. Thus by (8), $(L\cdot D)^{2}\geq L^{2}\cdot D^{2}\geq 5$, so $L\cdot D>2$. It follows that $L\cdot D\geq 3$. Hence, $L\cdot D-2\geq 1$. This inequality combining with (6) and (7) yields

[TABLE]

This implies $L\cdot D\leq 4$. The only possibilities are $L\cdot D=3$ or $L\cdot D=4$.

• •

If $D^{2}=1$ then $L\cdot D\leq 3$ by (6). Since $L\cdot D$ can only be either $3$ or $4$, $L\cdot D=3$ in this case. Furthermore, suppose that $L^{2}\geq 10$. If $L\cdot D=3$ and $D^{2}=1$ then $9=(L\cdot D)^{2}<10\leq L^{2}\cdot D^{2}$, a contradiction to (8).

• •

Now assume that $D^{2}\geq 2$. If $L\cdot D=3$, then $L^{2}\geq 7$ by (7), and $L^{2}\cdot D^{2}\geq 7\cdot 2=14>9=(L\cdot D)^{2}$, a contradiction to (8). Now assume that $L\cdot D=4$. Then the polygon $P_{L}$ associated to $L$ has at least $5$ vertices and one of its edges has lattice length $4$ by Lemma 2.1. Hence, $L^{2}\geq 9$ by Lemma 3.3. It follows that $16=(L\cdot D)^{2}<18\leq L^{2}\cdot D^{2}$, a contradiction to (8).

The proposition follows. ∎

5. Some Applications

The following corollary gives an affirmative answer for a stronger form of Fujita’s conjecture (Conjecture 1.1) in case of smooth complete toric surfaces. Note that for $n$-dimensional toric varieties, the Fujita’s conjecture is in fact a corollary of [4, Corollary 0.2] and [13, Theorem 1].

Corollary 5.1** ([4, 13]).**

Let $X$ be a smooth complete surface not isomorphic to $\mathbb{P}^{2}$. Let $L$ be an ample line bundle on $X$ such that $L\cdot C\geq 2$ for all toric invariant curve $C\subset X$. Then $\mathcal{O}_{X}(K_{X}+L)$ is globally generated. If $L^{2}\geq 10$ and $L\cdot C\geq 3$ for all toric invariant curve $C\subset X$, then $\mathcal{O}_{X}(K_{X}+L)$ is very ample.

Proof.

Suppose that $\mathcal{O}_{X}(K_{X}+L)$ is not globally generated. By Proposition 1.2, there exists a toric invariant curve $C$ such that $L\cdot C=0$ or $L\cdot C=1$, a contradiction. ∎

As a corollary, we have a stronger form of [8, Corollary 2.7] for smooth toric surfaces as follows.

Corollary 5.2**.**

If $A$ is an ample line bundle on a smooth complete toric surface $X$ not isomorphic to $\mathbb{P}^{2}$, then $|K_{X}+2A|$ is nef, and $|K_{X}+4A|$ is very ample.

Proof.

Take $L=2A$, then for any toric invariant curve $C\subset X$, $L\cdot C=2A\cdot C\geq 2$. By Proposition 1.2, $|K_{X}+2A|$ is nef. Similarly, take $L^{\prime}=4A$, then $(L^{\prime})^{2}=16A^{2}>10$, and $L\cdot C=4A\cdot C\geq 4$. By Proposition 1.2, $|K_{X}+4A|$ is very ample. ∎

Remark 5.3*.*

It would be interesting to see if we can apply the classification in Proposition 1.2 to the study of Iskovskikh-Shokurov conjecture [6] for conic bundles over smooth toric surfaces.