Toric log del Pezzo surfaces with one singularity

TL;DR

This paper classifies all toric log Del Pezzo surfaces with a single singularity and describes their embeddings into projective spaces as intersections of quadrics.

Contribution

It provides a complete classification of these surfaces and details their explicit embeddings, advancing understanding of their geometric structure.

Findings

Classification of all such surfaces achieved

Explicit descriptions of embeddings into projective spaces

Identification of the intersection of quadrics as embeddings

Abstract

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.

Figures

Equations

$${\mathcal{O}}_X(-m{K}_X)\cong {\Phi }_{|-m{K}_X|}^{\ast }\left({\mathcal{O}}_{\mathbb{P}(|-m{K}_X|)}\left(1\right)\right),$$

$${\mathbb{F}}_{\kappa }:=\left\{(\left[z_0:z_1:z_2\right],\left[t_1:t_2\right])\in {\mathbb{P}}_{\mathbb{C}}^2\times {\mathbb{P}}_{\mathbb{C}}^1|z_1{t}_1^{\kappa }=z_2{t}_2^{\kappa }\right\},\kappa \in {\mathbb{Z}}_{\ge 0},$$

$$\left\{\begin{array}[]{l}Q_{p}^{\left[1\right]}:=\text{\emph{conv}}\left(\left\{\tbinom{1}{-1},\tbinom{p}{1},\tbinom{-1}{0}\right\}\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ Q_{p}^{\left[2\right]}:=\text{\emph{conv}}\left(\left\{\tbinom{1}{-1},\tbinom{p}{1},\tbinom{p-1}{1},\tbinom{-1}{0}\right\}\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ Q_{p}^{\left[3\right]}:=\text{\emph{conv}}\left(\left\{\tbinom{1}{-1},\tbinom{p}{1},\tbinom{p-1}{1},\tbinom{-1}{0},\tbinom{0}{-1}\right\}\right)\end{array}\right\},$$

$$\rho \left(X_Q\right)=k⟺\exists p\in {\mathbb{Z}}_{>0}:X_Q\cong X_{Q_p^{\left[k\right]}},$$

$${\Phi }_{|-\ell K_{X_Q}|}:X_Q\hookrightarrow \mathbb{P}(\left|-\ell K_{X_Q}\right|)$$

$$p$$

$$0\le p<q\text{ and }\kappa c-\lambda d\equiv p\left(\text{mod }q\right),$$

$${\mathbf{n}}^{\prime }=p\mathbf{n}+q{\mathbf{n}}^{\prime \prime }\text{ and }\left\{\mathbf{n},{\mathbf{n}}^{\prime \prime }\right\}\text{ is a }\mathbb{Z}\text{-basis of }{\mathbb{Z}}^2.$$

$$\Psi \left(\sigma \right)={\mathbb{R}}_{\ge 0}\left(\genfrac{}{}{0ex}{}{1}{0}\right)+{\mathbb{R}}_{\ge 0}\left(\genfrac{}{}{0ex}{}{p}{q}\right).$$

$$q=\left|\det (\mathbf{n},{\mathbf{n}}^{\prime })\right|=2\text{ area}(T).$$

$$\frac{q}{2}=\text{area}(T)=♯(\text{int}(T)\cap {\mathbb{Z}}^2)+\frac{1}{2}♯(\partial (T)\cap {\mathbb{Z}}^2)-1,$$

$$♯(\partial (T)\cap {\mathbb{Z}}^2)\ge 3\Rightarrow ♯(\text{int}(T)\cap {\mathbb{Z}}^2)=0\text{ and necessarily }♯(\partial (T)\cap {\mathbb{Z}}^2)=3.$$

$$U_{\sigma }\cong {\mathbb{C}}^2/G=\text{Spec}(\mathbb{C}[z_1,z_2{]}^G),$$

$${\sigma }_i={\mathbb{R}}_{\ge 0}{\mathbf{n}}_i+{\mathbb{R}}_{\ge 0}{\mathbf{n}}_{i+1},i\in \{1,\dots ,\nu \},$$

$$\rho (X_{\Delta })=\nu -2,$$

$$I_{\Delta }:=\left\{i\in \{1,\dots ,\nu \}|q_i>1\right\},J_{\Delta }:=\left\{i\in \{1,\dots ,\nu \}|q_i=1\right\},$$

$$\text{Sing}(X_{\Delta })=\left\{\text{orb}({\sigma }_i)|i\in I_{\Delta }\right\}.$$

$$\frac{q_{i}}{q_{i}-p_{i}}=b_{1}^{(i)}-\frac{1}{b_{2}^{(i)}-\dfrac{\begin{array}[]{c}1\end{array}}{\begin{array}[]{cc}\ddots&\\ &b_{s_{i}-1}^{(i)}-\dfrac{1}{b_{s_{i}}^{(i)}}\end{array}}}\ \ ,$$

$${\mathbf{u}}_{j+1}^{(i)}=b_j^{(i)}{\mathbf{u}}_j^{(i)}-{\mathbf{u}}_{j-1}^{(i)},\forall j\in \{1,\dots ,s_i\}.$$

$$K_{X_{\Delta }}^2=12-\nu +\sum _{i\in I_{\Delta }}\left(\frac{q_i-p_i+1}{q_i}+\frac{q_i-{\hat{p}}_i+1}{q_i}-2+\sum _{j=1}^{s_i}\left(b_j^{(i)}-3\right)\right).$$

$$\widetilde{\Delta}:=\left\{\begin{array}[]{c}\text{ the cones }\left\{\left.\sigma_{i}\ \right|\ i\in J_{\Delta}\right\}\text{ and }\\ \left\{\left.\mathbb{R}_{\geq 0}\,\mathbf{u}_{j}^{(i)}+\mathbb{R}_{\geq 0}\,\mathbf{u}_{j+1}^{(i)}\ \right|\ i\in I_{\Delta},\ j\in\{0,1,\ldots,s_{i}\}\right\},\\ \text{together with their faces}\end{array}\right\}.$$

$$E^{(i)}:=\sum _{j=1}^{s_i}{E}_j^{(i)},i\in I_{\Delta },$$

$$E_j^{(i)}:=\overline{{\text{orb}}_{\tilde{\Delta }}({\mathbb{R}}_{\ge 0}{\mathbf{u}}_j^{(i)})}(\cong {\mathbb{P}}_{\mathbb{C}}^1),\forall j\in \{1,2,\dots ,s_i\},$$

$$r_{i}\mathbf{n}_{i}=\left\{\begin{array}[]{ll}\mathbf{u}_{s_{i-1}}^{(i-1)}+\mathbf{u}_{1}^{(i)},&\text{if }i\in I_{\Delta}^{\prime},\\ \mathbf{n}_{i-1}+\mathbf{u}_{1}^{(i)},&\text{if\emph{\ }}i\in I_{\Delta}^{\prime\prime},\\ \mathbf{u}_{s_{i-1}}^{(i-1)}+\mathbf{n}_{i+1},&\text{if }i\in J_{\Delta}^{\prime},\\ \mathbf{n}_{i-1}+\mathbf{n}_{i+1},&\text{if }i\in J_{\Delta}^{\prime\prime},\end{array}\right.$$

$$I_{\Delta }^{\prime }:=\left\{i\in I_{\Delta }|q_{i-1}>1\right\},I_{\Delta }^{\prime \prime }:=\left\{i\in I_{\Delta }|q_{i-1}=1\right\},$$

$$J_{\Delta }^{\prime }:=\left\{i\in J_{\Delta }|q_{i-1}>1\right\},J_{\Delta }^{\prime \prime }:=\left\{i\in J_{\Delta }|q_{i-1}=1\right\},$$

$$\left\{\text{Vertices of }\mathfrak{G}\right\}⟼\mathbb{Z},\left\{\text{Edges of }\mathfrak{G}\right\}⟼{\mathbb{Z}}^2,$$

$$\left\{\text{Vertices of }{\mathfrak{G}}_{\Delta }\right\}=\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_{\nu }\}\text{and }\left\{\text{Edges of }{\mathfrak{G}}_{\Delta }\right\}=\{\overline{{\mathbf{v}}_1{\mathbf{v}}_2},\dots ,\overline{{\mathbf{v}}_{\nu }{\mathbf{v}}_1}\},$$

Full text

1. Introduction

A smooth compact complex surface $X$ is called del Pezzo surface if its anticanonical divisor $-K_{X}$ is ample, i.e., if the rational map $\Phi_{\left|-mK_{X}\right|}:X\dashrightarrow\mathbb{P}(\left|-mK_{X}\right|)$ associated to the linear system $\left|-mK_{X}\right|$ becomes a closed embedding with

[TABLE]

for a suitable positive integer $m.$ (Pasquale del Pezzo [16] initiated the study of these surfaces in $1887.$) The degree $\deg(X)$ of a del Pezzo surface $X$ is defined to be the self-intersection number $(-K_{X})^{2}.$ The main classification result about these surfaces can be stated as follows (see [32, Theorem 24.4, pp. 119-121]):

Theorem 1.1**.**

Let $X$ be a del Pezzo surface of degree $d:=\deg(X).$ We have necessarily $1\leq d\leq 9,$ and $X$ is classified by $d$:

(i)* If $d=9,$ then $X$ is isomorphic to the projective plane $\mathbb{P}_{\mathbb{C}}^{2}.\vskip 3.0pt plus 1.0pt minus 1.0pt$

(ii) If $d=8,$ then $X$ is isomorphic either to $\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$ or to the blow-up of the projective plane $\mathbb{P}_{\mathbb{C}}^{2}$ at one point. 

(iii) If $1\leq d\leq 7,$ then $X$ is isomorphic to the blow-up of the projective plane $\mathbb{P}_{\mathbb{C}}^{2}$ at $9-d$ points.*

For $6\leq d\leq 9,$ such an $X$ is toric, i.e., it contains a $2$-dimensional algebraic torus $\mathbb{T}$ as a dense open subset, and is equipped with an algebraic action of $\mathbb{T}$ on $X$ which extends the natural action of $\mathbb{T}$ on itself. Taking into account the description of smooth compact toric surfaces by the ($\mathbb{Z}$-weighted) circular graphs (introduced in [37, Chapter I, §8], [38, pp. 42-46]), as well as [3, Proposition 6] and [41, Proposition 2.7], Oda expresses in [38, Proposition 2.21, pp. 88-89] this fact in the language of toric geometry as follows:

Theorem 1.2**.**

There exist five distinct toric del Pezzo surfaces up to isomorphism. They correspond to the circular graphs (with weights $-1,0,1$) shown in Figure 1. They are (i) $\mathbb{P}_{\mathbb{C}}^{2},$ (ii) $\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$ ($\cong\mathbb{F}_{0}$), (iii) the Hirzebruch surface $\mathbb{F}_{1},$ (iv) the equivariant blow-up of $\ \mathbb{P}_{\mathbb{C}}^{2}$ at two of the $\mathbb{T}$-fixed points, and (v) the equivariant blow-up of $\mathbb{P}_{\mathbb{C}}^{2}$ at the three $\mathbb{T}$-fixed points.

Note 1.3**.**

The so-called *Hirzebruch surfaces *(introduced in [26, §2])

[TABLE]

are toric. $\mathbb{F}_{\kappa}$ is usually identified with the total space $\mathbb{P}(\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}\oplus\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}(\kappa))$ of the $\mathbb{P}_{\mathbb{C}}^{1}$-bundle of degree $\kappa$ over $\mathbb{P}_{\mathbb{C}}^{1}.$ Furthermore, every smooth compact toric surface which has Picard number $2$ is necessarily isomorphic to a Hirzebruch surface (cf. [38, Corollary 1.29, p. 45]).

$\blacktriangleright$ The singular analogues. A normal compact complex surface $X$ with at worst log terminal singularities, i.e., quotient singularities, is called log del Pezzo surface if its anticanonical Weil divisor $-K_{X}$ is a $\mathbb{Q}$-Cartier ample divisor. The index of such an $X$ is defined to be the smallest positive integer $\ell$ for which $-\ell K_{X}$ is a Cartier divisor. The family of log del Pezzo surfaces of fixed index $\ell$ is known to be bounded. (See Nikulin [34], [35], [36], and Borisov [6, Theorem 2.1, p. 332].) Consequently, it seems to be rather interesting to classify log del Pezzo surfaces of given index $\ell$. This has been done for index $\ell=1$ by Hidaka & Watanabe [25] (by a direct generalization of Theorem 1.1) and Ye [42], and for index $\ell=2$ by Alexeev & Nikulin [1], [2] (in terms of diagrams of exceptional curves w.r.t. a suitable resolution of singularities). Related results are due to Kojima [31] (whenever the Picard number equals $1$) and Nakayama [33] (whose techniques apply even if one replaces $\mathbb{C}$ with an algebraically closed field of arbitrary characteristic). Based on Nakayama’s arguments, Fujita & Yasutake [22] succeeded recently to extend the classification even for $\ell=3.$ But for indices $\ell\geq 4$ the situation turns out to be much more complicated, and (apart from some partial results as those in [20], [21]) it is hard to expect a complete characterization of these surfaces in this degree of generality.

On the other hand, if we restrict our study to the subclass of toric log del Pezzo surfaces, the classification problem becomes considerably simpler: a) The only singularities which can occur are cyclic quotient singularities. b) To classify (not necessarily smooth) compact toric surfaces up to isomorphism it is enough to use the graph-theoretic method proposed in [12, §5] (which generalizes Oda’s graphs mentioned above): Two compact toric surfaces are isomorphic to each other if and only if their vertex singly- and edge doubly-weighted circular graphs (wve2c-graphs, for short) are isomorphic (see below Theorem 3.3). A detailed examination of the number-theoretic properties of the weights of these graphs led to the classification of all toric log del Pezzo surfaces having Picard number $1$ and index $\ell\leq 3$ in [12, §6] and [13]. In fact, the purely combinatorial part of the classification problem can be further simplified because it can be reduced to the classification of the so-called LDP-polygons (introduced in [15]) up to unimodular transformation. For $\ell=1$ these are the sixteen reflexive polygons (which were discovered by Batyrev in the 1980’s). More recently, Kasprzyk, Kreuzer & Nill [28, §6] developed a particular algorithm by means of which one creates an LDP-polygon (for given $\ell\geq 2$) by fixing a “special” edge and following a prescribed successive addition of vertices, and produced in this way the long lists of all LDP-polygons for $\ell\leq 17.$ (An explicit study for each of these $15346$ LDP-polygons is available on the webpage [8].)

$\blacktriangleright$ Restrictions on the singularities. At this point let us mention some remarkable results concerning the singularities of log del Pezzo surfaces having Picard number $1$: Belousov proved in [4], [5] that each of these surfaces admits at most $4$ singularities, Kojima [30] described the nature of the exceptional divisors w.r.t. the minimal resolution of those possessing exactly one singularity, and Elagin [17] constructed certain (non-toric) surfaces of this kind (realized as hypersurfaces of degree $4n-2$ in $\mathbb{P}_{\mathbb{C}}^{3}(1,2,2n-1,4n-3)$), and proved the existence of full exceptional sets of coherent sheaves over them.

Obviously, the maximal number of the singularities of a toric log del Pezzo surface equals the number of the edges of the corresponding LDP-polygon. (For an upper bound of this number see [15, Lemma 3.1].) In the present paper we classify all toric log del Pezzo surfaces with exactly one singularity (without laying a priori any restrictions on the Picard number or on the index) up to isomorphism.

Theorem 1.4**.**

Let $X_{Q}$ be a toric log del Pezzo surface (associated to an LDP-polygon $Q$) with exactly one singularity. Then the following hold true:*

(i) The Picard number $\rho\left(X_{Q}\right)$ of $X_{Q}$ can take only the values $1,2$ and $3.\vskip 3.0pt plus 1.0pt minus 1.0pt$

*(ii) If we define for every positive integer $p$ the LDP-polygons

[TABLE]

then for $k\in\{1,2,3\}$ we have

[TABLE]

and the wve2*c-graphs $\mathfrak{G}_{\Delta_{Q_{p}^{\left[k\right]}}}$ are those depicted in Figure 2. 

(iii) $X_{Q_{p}^{\left[1\right]}}$ is isomorphic to the weighted projective plane $\mathbb{P}_{\mathbb{C}}^{2}(1,1,p+1)$ and is obtained by contracting the $\infty$-section $\mathbb{P}(\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}(p+1))$ of $\ \mathbb{F}_{p+1}$. The surface $X_{Q_{p}^{\left[2\right]}}$ is obtained by blowing up a Hirzebruch surface $\mathbb{F}_{p}$ at one $\mathbb{T}$-fixed point, and contracting afterwards its $\infty$-section. $X_{Q_{p}^{\left[3\right]}}$ is obtained by blowing up $X_{Q_{p}^{\left[2\right]}}$ at one non-singular $\mathbb{T}$-fixed point. 

(iv) If $X_{Q}$ has index $\ell\geq 1$ and Picard number $\rho\left(X_{Q}\right)=k\in\{1,2,3\},$ then for $\ell$ odd $\geq 3$ either $X_{Q}\cong X_{Q_{\ell-1}^{\left[k\right]}}$ or $X_{Q}\cong X_{Q_{2\ell-1}^{\left[k\right]}},$ whereas for $\ell\in\{1\}\cup 2\mathbb{Z}$ we have $X_{Q}\cong X_{Q_{2\ell-1}^{\left[k\right]}}.$*

$\blacktriangleright$ Equations defining closed embeddings. For every del Pezzo surface $X$ of degree $d$ with $3\leq d\leq 9$ the anticanonical divisor $-K_{X}$ is already very ample, and $\Phi_{\left|-K_{X}\right|}$ gives rise to a realization of $X$ as a subvariety of projective degree $d$ in $\mathbb{P}_{\mathbb{C}}^{d}.$ (For $d=1$ and $d=2,$ one has to work with $-3K_{X}$ and $-2K_{X}$ instead to obtain realizations of $X$ as a subvariety of degree $9,$ and of degree $8$ in $\mathbb{P}_{\mathbb{C}}^{6},$ respectively.) Generalizations of these (or similar but more “economic”) embeddings of log del Pezzo surfaces of index $1$ and $2$ (in appropriate projective or weighted projective spaces) appear in [25] and [27]. Since every ample divisor on a compact toric surface is very ample (cf. [19] or [11, Corollary 2.2.19 (b), p. 71, and Proposition 6.1.10, pp. 269-270 ]), the map $\Phi_{\left|-\ell K_{X_{Q}}\right|}$ associated to the linear system $\left|-\ell K_{X_{Q}}\right|$ on a toric log del Pezzo surface $X_{Q}$ of index $\ell$ becomes a closed embedding. Koelman’s Theorem [29] and standard lattice point enumeration techniques enable us to describe $\Phi_{\left|-\ell K_{X_{Q}}\right|}(X_{Q})$ for those $X_{Q}$’s classified in Theorem 1.4 as follows:

Theorem 1.5**.**

Let $X_{Q}$ be a toric log del Pezzo surface of index $\ell\geq 1$ with exactly one singularity. Then the image of $X_{Q}\cong X_{Q_{p}^{\left[k\right]}}$ under the closed embedding

[TABLE]

is isomorphic to a subvariety of $\mathbb{P}_{\mathbb{C}}^{\delta_{Q_{p}^{\left[k\right]}}}$ of projective degree $d_{Q_{p}^{\left[k\right]}}$ which can be expressed as intersection of finitely many quadrics; $\delta_{Q_{p}^{\left[k\right]}}$ and $d_{Q_{p}^{\left[k\right]}}$ are given in the table:**

[TABLE]

On the other hand, the cardinality $\beta_{Q_{p}^{\left[k\right]}}$ of any minimal system of quadrics (generating the ideal which determines this subvariety) equals

[TABLE]

and the sectional genus $g_{Q_{p}^{\left[k\right]}}\emph{\vskip 6.0pt plus 2.0pt minus 2.0pt}$ of $X_{Q_{p}^{\left[k\right]}}$ equals

[TABLE]

The paper is organized as follows: In §2 we focus on the two non-negative, relatively prime integers $p=p_{\sigma}$ and $q=q_{\sigma}$ parametrizing the $2$-dimensional, rational, strongly convex polyhedral cones $\sigma,$ and explain how they characterize the $2$-dimensional toric singularities. In §3-§4 we recall some auxiliary geometric properties of compact toric surfaces and of those which are log del Pezzo. The proofs of Theorems 1.4 and 1.5 are given in sections 5 and 6, respectively. We use tools only from discrete and classical toric geometry, adopting the standard terminology from [11], [18], [23], and [38] (and mostly the notation introduced in [12]).

2. Two-dimensional toric singularities

Let $\sigma=\mathbb{R}_{\geq 0}\mathbf{n}+\mathbb{R}_{\geq 0}\mathbf{n}^{\prime}\subset\mathbb{R}^{2}$ be a 2-dimensional, rational, strongly convex polyhedral cone. Without loss of generality we may assume that $\mathbf{n}=\tbinom{a}{b}$, $\mathbf{n}^{\prime}=\tbinom{c}{d}\in\mathbb{Z}^{2},$ and that both $\mathbf{n}$ and $\mathbf{n}^{\prime}$ are primitive elements of $\mathbb{Z}^{2}$, i.e., gcd$\left(a,b\right)=1$ and gcd$\left(c,d\right)=1.$

Lemma 2.1**.**

Consider $\kappa,\lambda\in\mathbb{Z},$ such that $\kappa a-\lambda b=1.$ If $q:=\left|ad-bc\right|,$ and $p$ is the unique integer with

[TABLE]

then gcd*$\left(p,q\right)=1$, and there exists a primitive element $\mathbf{n}^{\prime\prime}\in\mathbb{Z}^{2}$ such that*

[TABLE]

Moreover, there is a unimodular transformation $\Psi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},$ $\Psi\left(\mathbf{x}\right):=\Xi\,\mathbf{x,}$ with $\Xi\in$ GL${}_{2}(\mathbb{Z}),$ such that

[TABLE]

Proof.

See [13, Lemma 2.1 and Lemma 2.2].

Henceforth, we call $\sigma$ a $(p,q)$-cone. By $U_{\sigma}:=$ Spec$(\mathbb{C}[\sigma^{\vee}\cap\mathbb{Z}^{2}])$ we denote the affine toric variety associated to $\sigma$ (by means of the monoid $\sigma^{\vee}\cap\mathbb{Z}^{2}$, where $\sigma^{\vee}$ is the dual of $\sigma$) and by orb$(\sigma)$ the single point being fixed under the usual action of the algebraic torus $\mathbb{T}:=$ Hom${}_{\mathbb{Z}}(\mathbb{Z}^{2},\mathbb{C}^{\ast})$ on $U_{\sigma}.$

Proposition 2.2**.**

*The following conditions are equivalent: 

(i) $\left\{\mathbf{n},\mathbf{n}^{\prime}\right\}$ is a $\mathbb{Z}$-basis of $\mathbb{Z}^{2}$.  

(ii) $q=1$ (and consequently, $p=0$).

(iii) conv$(\{\mathbf{0},\mathbf{n},\mathbf{n}^{\prime}\})\cap\mathbb{Z}^{2}=\{\mathbf{0},\mathbf{n},\mathbf{n}^{\prime}\}.$ (“conv” *is abbreviation for convex hull.)

(iv)* $U_{\sigma}\cong\mathbb{C}^{2}.$*

Proof.

Let $T$ be the triangle conv$(\{\mathbf{0},\mathbf{n},\mathbf{n}^{\prime}\}).$ The implication (i)$\Rightarrow$(ii) is obvious because

[TABLE]

By Pick’s formula (cf. [23, p. 113]) we obtain

[TABLE]

where “int” and $\partial$ are abbreviations for interior and boundary, respectively. If $q=1,$ then

[TABLE]

Hence, (ii)$\Rightarrow$(iii) is also true. (iii)$\Rightarrow$(i) follows from [24, Theorem 4, p. 20]. For the proof of the equivalence of conditions (i) and (iv) see [38, Theorem 1.10, p. 15].

If the conditions of Proposition 2.2 are satisfied, then $\sigma$ is said to be a basic cone. On the other hand, whenever $q>1$ we have the following:

Proposition 2.3**.**

$\emph{orb}(\sigma)\in U_{\sigma}$* is a cyclic quotient singularity. In particular,*

[TABLE]

with $G\subset$ GL$\left(2,\mathbb{C}\right)$ denoting the cyclic group $G$ of order $q$ which is generated by diag$(\zeta_{q}^{-p},\zeta_{q})$ ($\zeta_{q}:=$ exp$(2\pi\sqrt{-1}/q)$)* and acts on $\mathbb{C}^{2}=$ Spec$(\mathbb{C}[z_{1},z_{2}])$ linearly and effectively.*

Proof.

See [11, Proposition 10.1.2, pp. 460-461], [23, § 2.2, pp. 32-34] and [38, Proposition 1.24, p.30].

By Proposition 2.4 these two numbers $p=p_{\sigma}$ and $\ q=q_{\sigma}$ parametrize uniquely the isomorphism class of the germ* $(U_{\sigma}$, orb$\left(\sigma\right))$, up to replacement of *$p$ by its socius $\widehat{p}$ (which corresponds just to the interchange of the coordinates). [The socius $\widehat{p}$ of $p$ is defined to be the uniquely determined integer, so that $0\leq\widehat{p}<q\emph{,}$ gcd$(\widehat{p},q)=1,$ and $p\,\widehat{p}\equiv 1$(mod $q$).]

Proposition 2.4**.**

Let $\sigma,\tau\subset\mathbb{R}^{2}$ be two $2$-dimensional, rational, stronly convex polyhedral cones. Then the following conditions are equivalent:*

*(i)  *There is a $\mathbb{T}$-equivariant isomorphism $U_{\sigma}\cong U_{\tau}$ mapping orb$\left(\sigma\right)$ onto orb$\left(\tau\right)$.

(ii) There is a unimodular transformation $\Psi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},$ $\Psi\left(\mathbf{x}\right):=\Xi\,\mathbf{x,}$ $\Xi\in$ GL${}_{2}(\mathbb{Z}),$ such that* *$\Psi\left(\sigma\right)=\tau.\vskip 3.0pt plus 1.0pt minus 1.0pt$

(iii) For the numbers $p_{\sigma},$ $p_{\tau},$ $q_{\sigma},$ $q_{\tau}$ associated to $\sigma,\tau$ *(by Lemma 2.1) we have $q_{\tau}=q_{\sigma}$ and either $p_{\tau}=p_{\sigma}$ or $p_{\tau}=\widehat{p}_{\sigma}.$

Proof.

See [13, Proposition 2.4].

3. Compact toric surfaces

Every compact toric surface is a $2$-dimensional toric variety $X_{\Delta}$ associated to a complete fan $\Delta$ in $\mathbb{R}^{2},$ i.e., a fan having $2$-dimensional cones as maximal cones and whose support $\left|\Delta\right|$ is the entire $\mathbb{R}^{2}$ (see [38, Theorem 1.11, p. 16]). Consider a complete fan $\Delta$ in $\mathbb{R}^{2}$ and suppose that

[TABLE]

are its $2$-dimensional cones (with $\nu\geq 3$ and $\mathbf{n}_{i}\in\mathbb{Z}^{2}$ primitive for all $i\in\{1,\ldots,\nu\}$), enumerated in such a way that $\mathbf{n}_{1},\ldots,\mathbf{n}_{\nu}$ go anticlockwise around the origin exactly once in this order (under the usual convention: $\mathbf{n}_{\nu+1}:=\mathbf{n}_{1},$ $\mathbf{n}_{0}:=\mathbf{n}_{\nu}$). $X_{\Delta}$ is obtained by gluing the affine charts $U_{\sigma_{i}}$ along the open subsets which are defined by the rays $\sigma_{i}\cap\sigma_{i+1},$ for all $i\in\{1,\ldots,\nu\}$ (cf. [38, Theorem 1.4, p. 7]). Since $\Delta$ is simplicial, the Picard number $\rho(X_{\Delta})$ of $X_{\Delta}$ (i.e., the rank of its Picard group Pic$(X_{\Delta})$) equals

[TABLE]

(see [23, p. 65]). Now suppose that $\sigma_{i}$ is a $(p_{i},q_{i})$-cone for all $i\in\{1,\ldots,\nu\}$ and introduce the notation

[TABLE]

to separate the indices corresponding to non-basic from those corresponding to basic cones. By Propositions 2.2 and 2.3 the singular locus of $X_{\Delta}$ equals

[TABLE]

For all $i\in I_{\Delta}$ consider the negative-regular continued fraction expansion of

[TABLE]

and define $\mathbf{u}_{1}^{(i)}:=\mathbf{n}_{i},\ \mathbf{u}_{1}^{(i)}:=\frac{1}{q_{i}}((q_{i}-p_{i})\mathbf{n}_{i}+\mathbf{n}_{i+1}),$ and

[TABLE]

It is easy to see that $\mathbf{u}_{s_{i}+1}^{(i)}=\mathbf{n}_{i+1},$ and that $b_{j}^{(i)}$ are integers $\geq 2,$ for all indices $j\in\{1,\ldots,s_{i}\}.$ According to [12, Proposition 4.9, p. 99], the self-intersection number of the canonical divisor $K_{X_{\Delta}}$ of $X_{\Delta}$ equals

[TABLE]

By construction, the birational morphism $f:X_{\widetilde{\Delta}}\longrightarrow X_{\Delta}$ induced by the refinement

[TABLE]

of the fan $\Delta$ is the* minimal desingularization* of $X_{\Delta}.$ The exceptional divisor

[TABLE]

replacing orb$(\sigma_{i})$ via $f$ has

[TABLE]

(i.e., the closures of the orbits of the new rays w.r.t. $\widetilde{\Delta}$) as its components, and self-intersection number $(E_{j}^{(i)})^{2}=-b_{j}^{(i)}.$ Moreover, $\overline{C}_{i}:=\overline{\text{orb}_{\widetilde{\Delta}}(\mathbb{R}_{\geq 0}\,\mathbf{n}_{i})}$ is the *strict transform *of $C_{i}:=\overline{\text{orb}_{\Delta}(\mathbb{R}_{\geq 0}\,\mathbf{n}_{i})}$ w.r.t. $f$ for all $i\in\{1,2,\ldots,\nu\}.$

Definition 3.1**.**

For every $i\in\{1,\ldots,\nu\}$ we introduce integers $r_{i}$ uniquely determined by the conditions:

[TABLE]

where

[TABLE]

and

[TABLE]

with $I_{\Delta},J_{\Delta}$ as in (3.3).

By [12, Lemma 4.3], for $i\in\{1,\ldots,\nu\},$ $-r_{i}$ is nothing but the self-intersection number $\overline{C}_{i}^{2}$ of * $\overline{C}_{i}.$* The triples $(p_{i},q_{i},r_{i}),$ $i\in\{1,2,\ldots,\nu\},$ are used to define the wve2c-graph* $\mathfrak{G}_{\Delta}.$*

Definition 3.2**.**

A circular graph is a plane graph whose vertices are points on a circle and whose edges are the corresponding arcs (on this circle, each of which connects two consecutive vertices). We say that a circular graph $\mathfrak{G}$ is $\mathbb{Z}$-weighted at its vertices and double $\mathbb{Z}$-weighted at its edges (and call it wve2c-graph, for short) if it is accompanied by two maps

[TABLE]

assigning to each vertex an integer and to each edge a pair of integers, respectively. To every complete fan $\Delta$ in $\mathbb{R}^{2}$ (as described above) we associate an anticlockwise directed wve2c-graph $\mathfrak{G}_{\Delta}$ with

[TABLE]

$(\mathbf{v}_{\nu+1}:=\mathbf{v}_{1}),$ by defining its “weights” as follows:

[TABLE]

The reverse graph $\mathfrak{G}_{\Delta}^{\text{rev}}$ of $\mathfrak{G}_{\Delta}$ is the directed wve2c-graph which is obtained by changing the double weight $\left(p_{i},q_{i}\right)$ of the edge $\overline{\mathbf{v}_{i}\mathbf{v}_{i+1}}$ into $(\widehat{p}_{i},q_{i})$ and reversing the initial anticlockwise direction of $\mathfrak{G}_{\Delta}$ into clockwise direction (see Figure 3).

Theorem 3.3**.**

Let $\Delta,$ $\Delta^{\prime}$ be two complete fans in $\mathbb{R}^{2}.$ Then the following conditions are equivalent: *

(i) The compact toric surfaces $X_{\Delta}$ and $X_{\Delta^{\prime}}$ are isomorphic. 

(ii) Either $\mathfrak{G}_{\Delta^{\prime}}\overset{\text{\emph{gr.}}}{\cong}\mathfrak{G}_{\Delta}$ or $\mathfrak{G}_{\Delta^{\prime}}\overset{\text{\emph{gr.}}}{\cong}\mathfrak{G}_{\Delta}^{\text{\emph{rev}}}.$*

Here “$\,\overset{\text{gr.}}{\cong}\,$” indicates graph-theoretic isomorphism (i.e., a bijection between the sets of vertices which preserves the corresponding weights). For further details and for the proof of Theorem 3.3 (which can be viewed as an appropriate generalization of Proposition 2.4 for complete fans in $\mathbb{R}^{2}$) the reader is referred to [12, §5]. [Convention: To be absolutely compatible with Oda’s circular graphs we omit the weights of the edges which are equal to $(0,1)$, i.e., those corresponding to basic cones, whenever we draw a wve2c-graph.]

4. Toric log del Pezzo surfaces and LDP-polygons

Definition 4.1**.**

Let $Q\subset\mathbb{R}^{2}$ be a convex polygon. Denote by $\mathcal{V}(Q)$ and $\mathcal{F}(Q)$ the set of its vertices and the set of its facets (edges), respectively. $Q$ is called an LDP-polygon if it contains the origin in its interior, and its vertices belong to $\mathbb{Z}^{2}$ and are primitive. (Obviously, the image of an LDP-polygon under a unimodular transformation is again an LDP-polygon.)

If $Q$ is an LDP-polygon, we shall denote by $X_{Q}$ the compact toric surface $X_{\Delta_{Q}}$ constructed by means of the fan

[TABLE]

where $\sigma_{F}:=\left\{\left.\lambda\mathbf{x}\,\right|\,\mathbf{x}\in F\text{ and }\lambda\in\mathbb{R}_{\geq 0}\right\}$ for all $F\in\mathcal{F}(Q).$

Proposition 4.2**.**

(i)* A compact toric surface is log del Pezzo if and only if it is isomorphic to $X_{Q}$ for some LDP-polygon $Q.\vskip 3.0pt plus 1.0pt minus 1.0pt$

(ii) There is a one-to-one correspondence*

[TABLE]

Proof.

(i) This follows from [12, Remark 6.7, p. 107]. 

(ii) If $Q$ is an LDP-polygon, $\Psi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},$ $\Psi\left(\mathbf{x}\right):=\Xi\,\mathbf{x,}$ $\Xi\in$ GL${}_{2}(\mathbb{Z}),$ a unimodular transformation, and $Q^{\prime}:=\Psi(Q),$ then

[TABLE]

By Theorem 3.3, $X_{Q}$ and $X_{Q^{\prime}}$ are isomorphic. And conversely, if $X_{Q}$ and $X_{Q^{\prime}}$ are isomorphic for some LDP-polygons $Q,Q^{\prime},$ then

[TABLE]

Thus, by (4.1) there exists an automorphism $\varpi$ of the lattice $\mathbb{Z}^{2}=\mathbb{Z}\tbinom{1}{0}\oplus\mathbb{Z}\tbinom{0}{1}$ with

[TABLE]

such that $\varpi_{\mathbb{R}}(\Delta_{Q})=\Delta_{Q^{\prime}}$ (preserving/reversing the ordering of the cones), where

[TABLE]

denotes its scalar extension. Obviously, $\varpi_{\mathbb{R}}(Q)=Q^{\prime}.$

Note 4.3**.**

Let $Q$ be an arbitrary LDP-polygon. For each $F\in\mathcal{F}(Q)$ assume that $\sigma_{F}$ is a $(p_{F},q_{F})$-cone. Then from [12, Lemma 6.8] one concludes that the index $\ell$ of $X_{Q}$ equals

[TABLE]

If we consider the polar polygon $\mathring{Q}:=\left\{\left.\mathbf{y}\in\text{Hom}_{\mathbb{R}}(\mathbb{R}^{2},\mathbb{R})\right|\left\langle\mathbf{y},\mathbf{x}\right\rangle\geq-1,\ \forall\mathbf{x}\in Q\right\}$ of $Q,$ where $\left\langle\cdot,\cdot\right\rangle:$ Hom${}_{\mathbb{R}}(\mathbb{R}^{2},\mathbb{R})\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ denotes the usual inner product, then $\mathring{Q}$ contains the origin in its interior, and the index $\ell$ of $X_{Q}$ equals

[TABLE]

Moreover, if $F\in\mathcal{F}(Q),$ denoting by $\boldsymbol{\eta}_{F}$ the unique primitive $\boldsymbol{\eta}_{F}\in\mathbb{Z}^{2}$ for which $\left\langle\boldsymbol{\eta}_{F},\mathbf{x}\right\rangle=l_{F},\ \forall\mathbf{x}\in F,$ we have

[TABLE]

5. Proof of the classification theorem 1.4

Let $Q$ be an LDP-polygon with vertex set $\mathcal{V}(Q)=\left\{\mathbf{n}_{1},\ldots,\mathbf{n}_{\nu}\right\},$ $\nu\geq 3.$ Assume that $\sigma_{i},$ $i\in\{1,\ldots,\nu\},$ are the $2$-dimensional cones of $\Delta_{Q},$ defined and ordered (anticlockwise) as in (3.1), and that only one of these cones, say $\sigma_{1},$ is a non-basic $(p,q)$-cone (i.e., $q>1$). By Lemma 2.1, there is a unimodular transformation $\Psi_{1}:\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2},$ $\Psi_{1}\left(\mathbf{x}\right):=\Xi\,\mathbf{x,}$ $\Xi\in$ GL${}_{2}(\mathbb{Z}),$ such that

[TABLE]

Without loss of generality we may assume that $\det(\Xi)=1$ (because otherwise the proof of Theorem 1.4 which follows can be performed similarly if one works with the vertices ordered clockwise). This means that $\Psi_{1}(\mathbf{n}_{1})=\tbinom{1}{0}$ and $\Psi_{1}(\mathbf{n}_{2})=\tbinom{p}{q}.$ We set $\mathbf{w}_{i}:=\Psi_{1}(\mathbf{n}_{i}),$ for all $i\in\{1,\ldots,\nu\}$ (and $\mathbf{w}_{\nu+1}:=\mathbf{w}_{1}$). Since all cones of $\Delta_{\Psi_{1}\left(Q\right)}$ are strongly convex and $\left|\Delta_{\Psi_{1}\left(Q\right)}\right|=\mathbb{R}^{2}$,

[TABLE]

Lemma 5.1**.**

(i)* The cones $\mathbb{R}_{\geq 0}\mathbf{w}_{\mu}+\mathbb{R}_{\geq 0}\mathbf{w}_{1}$ and $\mathbb{R}_{\geq 0}\mathbf{w}_{2}+\mathbb{R}_{\geq 0}\mathbf{w}_{\mu}$ are basic. 

(ii) $q=p+1$ (and consequently, $\widehat{p}=p$ and $\mathbf{w}_{\mu}=\tbinom{-1}{-1}$).*

Proof.

(i) Using Proposition 2.2 it suffices to prove that

[TABLE]

Obviously, $\mathcal{V}(\Psi_{1}\left(Q\right))\mathbb{r}\{\mathbf{w}_{\mu},\mathbf{w}_{1},\mathbf{w}_{2}\}$ is either empty or a subset of $(\mathcal{U}_{1}\cup\mathcal{U}_{2})\cap\mathbb{Z}^{2},$ where

[TABLE]

and

[TABLE]

($\left\{\left.\tbinom{x}{y}\in\mathbb{R}^{2}\right|qx-(p-1)y=q\right\}$ is the supporting line of the edge conv$(\{\mathbf{w}_{1},\mathbf{w}_{2}\})$ of $\Psi_{1}\left(Q\right).$) If conv$(\{\mathbf{w}_{\mu},\mathbf{w}_{1}\})\in\mathcal{F}(\Psi_{1}\left(Q\right)),$ i.e., if $\mu=\nu,$ the first equality in (5.2) is obvious (because $\Psi_{1}\left(\sigma_{\nu}\right)$ is basic by definition). If conv$(\{\mathbf{w}_{\mu},\mathbf{w}_{1}\})\notin\mathcal{F}(\Psi_{1}\left(Q\right)),$ then $\mathcal{V}(\Psi_{1}\left(Q\right))\cap\mathcal{U}_{1}\neq\varnothing,$ and if we would assume that

[TABLE]

then there would be a

[TABLE]

leading to contradiction (because $\Psi_{1}\left(\sigma_{\xi-1}\right)$ is basic by definition). Similar arguments (using $\mathcal{U}_{2}$ instead of $\mathcal{U}_{1}$) show that the second equality in (5.2) is also true. 

(ii) By (i), $\left|\det(\mathbf{w}_{\mu},\mathbf{w}_{1})\right|=\left|\det(\mathbf{w}_{2},\mathbf{w}_{\mu})\right|=1,$ i.e., $b\in\{\pm 1\},$ and one of the following conditions is satisfied:

[TABLE]

(5.4) gives $a=\tfrac{1+p}{q}>0$ which is not true (because $a<0$ by (5.1)). Similarly, (5.6) is not true (as we would have $a=\tfrac{p-1}{q}\geq 0$ ). In case (5.8),

[TABLE]

Therefore, (5.10) is necessarily true and $a=-\tfrac{p+1}{q}$. Now since $q\mid p+1$ and $p<q,$ we have

[TABLE]

and $p+1\mid\left(p^{2}-1\right)\Rightarrow\widehat{p}=p.$

Lemma 5.2**.**

There exists a unimodular transformation $\Psi_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ such that

[TABLE]

with $\Psi_{2}\tbinom{1}{0}=\tbinom{1}{-1},$ $\Psi_{2}\tbinom{p}{p+1}=\tbinom{p}{1},$ and $\Psi_{2}\tbinom{-1}{-1}=\tbinom{-1}{0}.$

Proof.

It is enough to define \Psi_{2}(\mathbf{x}):=\left(\begin{array}[]{rc}\QATOPD..{1}{-1}&\QATOPD..{0}{1}\end{array}\right)\mathbf{x},\ \forall\mathbf{x}\in\mathbb{R}^{2}.

Next, we set $\Upsilon:=\Psi_{2}\circ\Psi_{1}$, $\mathbf{v}_{i}:=\Upsilon(\mathbf{n}_{i}),$ for all $i\in\{1,\ldots,\nu\}$ (and $\mathbf{v}_{\nu+1}:=\mathbf{v}_{1}$). Starting with the minimal generators $\mathbf{v}_{1}=\tbinom{1}{-1},$ $\mathbf{v}_{2}=\tbinom{p}{1}$ of the unique non-basic cone $\Upsilon\left(\sigma_{1}\right)$ of $\Delta_{\Upsilon\left(Q\right)},$ and with $\mathbf{v}_{\mu}=\tbinom{-1}{0}\in\mathcal{V}(\Upsilon\left(Q\right)),$ we shall study the restrictions on the location of the remaining vertices of $\Upsilon\left(Q\right)$ in detail.

Lemma 5.3**.**

There is no convex polygon having three collinear vertices.

Proof.

This is due to the fact that the vertices of a convex polygon are its extreme points. (See, e.g., [7, p. 30 and p. 45].)

Lemma 5.4**.**

The LDP-polygon $\Upsilon\left(Q\right)$ (with $\mathcal{V}(\Upsilon\left(Q\right))=\left\{\mathbf{v}_{1},\ldots,\mathbf{v}_{\nu}\right\}$)* has the following properties*:*

(i) Setting $k:=\nu-2,$ we have necessarily $k\in\{1,2,3\}.$ Moreover, $\Upsilon\left(Q\right)=Q_{p}^{\left[k\right]}$ for $k\in\{1,3\},$ and either $\Upsilon\left(Q\right)=Q_{p}^{\left[2\right]}$ or $\Upsilon\left(Q\right)=\check{Q}_{p}^{\left[2\right]}$ for $k=2,$ where $Q_{p}^{\left[1\right]},Q_{p}^{\left[2\right]},Q_{p}^{\left[3\right]}$ are the polygons defined in (1.1), and*

[TABLE]

(ii)* $Q_{p}^{\left[2\right]}$ and $\check{Q}_{p}^{\left[2\right]}$ are lattice-equivalent.*

Proof.

(i) If $\mathcal{U}_{1}^{\prime}:=\left\{\tbinom{x}{y}\in\Psi_{2}(\mathcal{U}_{1})\left|y\leq-2\right.\right\},$ we claim that $\mathcal{U}_{1}^{\prime}\cap\mathcal{V}(\Upsilon\left(Q\right))=\varnothing.$ If $\mathbf{v}_{\mu+1}\in\mathcal{U}_{1}^{\prime}\cap\mathcal{V}(\Upsilon\left(Q\right)),$ then we would have $\left|\det(\mathbf{v}_{\mu},\mathbf{v}_{\mu+1})\right|\geq 2,$ contradicting to the basicness of the cone $\Upsilon\left(\sigma_{\mu}\right).$ If

[TABLE]

then we would again have $\left|\det(\mathbf{v}_{\mu+1},\mathbf{v}_{\mu+2})\right|\geq 2,$ contradicting to the basicness of the cone $\Upsilon\left(\sigma_{\mu+1}\right).$ Repeating successively this procedure (until we arrive at $\mathbf{v}_{\nu}$) we bear out our assertion, as well as the implication

[TABLE]

Correspondingly, for $\mathcal{U}_{2}^{\prime}:=\left\{\tbinom{x}{y}\in\Psi_{2}(\mathcal{U}_{2})\left|y\geq 2\right.\right\}$ we show that $\mathcal{U}_{2}^{\prime}\cap\mathcal{V}(\Upsilon\left(Q\right))=\varnothing,$ and

[TABLE]

Hence, $\mathcal{V}(\Upsilon\left(Q\right))\mathbb{r}\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{\mu}\}$ is either empty or a subset of

[TABLE]

Taking into account Lemma 5.3 we conclude that

[TABLE]

Therefore, $k\in\{1,2,3\}$ and there are only four possibilities:

$\bullet$ Case (a): If $k=1,$ then $\nu=\mu=3$ and $\Upsilon\left(Q\right)=\text{conv}(\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\})=Q_{p}^{\left[1\right]}.$

$\bullet$ Case (b): If $k=2,$ then $\nu=4$ and either $\Upsilon\left(Q\right)=Q_{p}^{\left[2\right]},$ $\mu=4,\,$or $\Upsilon\left(Q\right)=\check{Q}_{p}^{\left[2\right]},$ $\mu=3.\vskip 3.0pt plus 1.0pt minus 1.0pt$

$\bullet$ Case (c): If $k=3,$ then $\nu=5,$ $\mu=4$ and $\Upsilon\left(Q\right)=Q_{p}^{\left[3\right]}.\vskip 3.0pt plus 1.0pt minus 1.0pt$

(ii) $Q_{p}^{\left[2\right]}$ is mapped onto $\check{Q}_{p}^{\left[2\right]}$ under the unimodular transformation

[TABLE]

and $\mathfrak{Y}(\mathbf{v}_{1})=\mathbf{v}_{2},$ $\mathfrak{Y}(\mathbf{v}_{2})=\mathbf{v}_{1},\mathfrak{\ Y}\tbinom{-1}{0}=\tbinom{-1}{0},$ $\mathfrak{Y}\tbinom{p-1}{1}=\tbinom{0}{-1}.$

Note 5.5**.**

The set conv$(\{\mathbf{v}_{1},\mathbf{v}_{2}\})\cap\mathbb{Z}^{2}$ is empty for $p$ even and consists of the single lattice point $\tbinom{\frac{1}{2}\vskip 3.0pt plus 1.0pt minus 1.0pt(p+1)}{0}$ for $p$ odd. Thus, the number of the lattice points belonging to the boundary of $Q_{p}^{\left[k\right]},$ $k\in\{1,2,3\},$ equals $k+2$ whenever $p$ is even and $k+3$ whenever $p$ is odd. Since area$(Q_{p}^{\left[k\right]})=\tfrac{p+k}{2}+1,$ Pick’s formula gives

[TABLE]

Proof of Theorem 1.4. (i)-(ii) Up to isomorphism, every toric log del Pezzo surface with exactly one singularity is of the form $X_{Q}$ with $Q$ as above. By (3.2), Lemma 5.4 and Proposition 4.2 we infer that the Picard number $\rho\left(X_{Q}\right)$ of $X_{Q}$ can take only the values $1,2$ and $3,$ and that for $k\in\{1,2,3\},$

[TABLE]

(Note that for $k=2,$ $\mathfrak{Y}$ induces a graph-theoretic isomorphism $\mathfrak{G}_{\Delta_{\check{Q}_{p}^{\left[2\right]}}}\overset{\text{gr.}}{\cong}\mathfrak{G}_{\Delta_{Q_{p}^{\left[2\right]}}}^{\text{rev}},$ meaning that $X_{Q_{p}^{\left[2\right]}}\cong X_{\check{Q}_{p}^{\left[2\right]}}.$) The fan $\widetilde{\Delta}_{Q_{p}^{\left[k\right]}}$ which is used to construct the minimal desingularization of $X_{Q_{p}^{\left[k\right]}}$ (as explained in §3) contains just one additional ray (compared with $\Delta_{Q_{p}^{\left[k\right]}}$), namely $\mathbb{R}_{\geq 0}\tbinom{1}{0}.$ The closure of its orbit constitutes the single exceptional divisor, say $E,$ w.r.t. this desingularization, with $E^{2}=-(p+1).$ Setting $\mathbf{u}_{E}:=\tbinom{1}{0}$ we compute the integers $r_{i},$ i\in\{1,\ldots,k+2\},\,\(defined in (3.5)) in the three different cases: 

$\bullet$ Case (a): If $k=1,$ then $\mathbf{v}_{1}=\tbinom{1}{-1},$ $\mathbf{v}_{2}=\tbinom{p}{1},\mathbf{v}_{3}=\tbinom{-1}{0},$ and

[TABLE]

$\bullet$ Case (b): If $k=2,$ then $\mathbf{v}_{1}=\tbinom{1}{-1},$ $\mathbf{v}_{2}=\tbinom{p}{1},$ $\mathbf{v}_{3}=\tbinom{p-1}{1},$ $\mathbf{v}_{4}=\tbinom{-1}{0},$ and

[TABLE]

$\bullet$ Case (c): If $k=3,$ then $\mathbf{v}_{1}=\tbinom{1}{-1},$ $\mathbf{v}_{2}=\tbinom{p}{1},$ $\mathbf{v}_{3}=\tbinom{p-1}{1},$ $\mathbf{v}_{4}=\tbinom{-1}{0},$ $\mathbf{v}_{5}=\tbinom{0}{-1},$ and

[TABLE]

Hence, the wve2c-graphs $\mathfrak{G}_{\Delta_{Q_{p}^{\left[k\right]}}}$ are indeed those depicted in Figure 2.

(iii) Defining for every positive integer $p$ the complete fan

[TABLE]

we see that $X_{\mathfrak{D}_{p}}\cong\mathbb{F}_{p+1},$ having $\overline{\text{orb}_{\mathfrak{D}_{p}}(\mathbb{R}_{\geq 0}\tbinom{1}{0})}$ as its $\infty$-section. The surfaces $X_{Q_{p}^{\left[k\right]}}$ are characterized as follows:

$\bullet$ Case (a): If $k=1,$ then $X_{Q_{p}^{\left[1\right]}}\cong$ $\mathbb{P}_{\mathbb{C}}^{2}(1,1,p+1)$ (see [13, Lemma 6.1]), and it is obtained by contracting the $\infty$-section of $X_{\mathfrak{D}_{p}}.$ In fact, since $X_{\mathfrak{D}_{p}}=X_{\widetilde{\Delta}_{Q_{p}^{\left[1\right]}}}$ is the minimal desingularization of $X_{Q_{p}^{\left[1\right]}},$ the surface $X_{Q_{p}^{\left[1\right]}}$ is nothing but the anticanonical model of $X_{\mathfrak{D}_{p}}$ (in the sense of Sakai [39]). 

$\bullet$ Case (b): If $k=2,$ the star subdivision of $\mathfrak{D}_{p-1}$ w.r.t the cone $\mathbb{R}_{\geq 0}\tbinom{1}{0}+\mathbb{R}_{\geq 0}\tbinom{p-1}{1}$ induces the equivariant blow-up $X_{\widetilde{\Delta}_{Q_{p}^{\left[2\right]}}}\longrightarrow$ $X_{\mathfrak{D}_{p-1}}$ with the orbit of this cone as centre (cf. [11, Proposition 3.3.15, p. 130], [37, Corollary 7.5, p. 45] or [18, Theorem VI.7.2, pp. 249-250]). Thus, the surface $X_{Q_{p}^{\left[2\right]}}$ is obtained by contracting the strict transform of the $\infty$-section of $X_{\mathfrak{D}_{p-1}}$ on $X_{\widetilde{\Delta}_{Q_{p}^{\left[2\right]}}}.$

$\bullet$ Case (c): If $k=3,$ we construct the surface $X_{Q_{p}^{\left[3\right]}}$ from $X_{Q_{p}^{\left[2\right]}}$ by using the equivariant birational morphism induced by the star subdivision of $\mathfrak{D}_{p-1}$ w.r.t the cone $\mathbb{R}_{\geq 0}\tbinom{-1}{0}+\mathbb{R}_{\geq 0}\tbinom{1}{-1}$, i.e., by blowing up its orbit (which is a non-singular $\mathbb{T}$-fixed point of $X_{Q_{p}^{\left[2\right]}}$).

Taking into account that we pass from $X_{\mathfrak{D}_{p-1}}$ to $X_{\mathfrak{D}_{p}}$ (and vice versa) by an elementary transformation (cf. [12, Remark 6.3, pp. 105-106]), we illustrate in Figure 4 how the equivariant birational morphisms connecting all the above mentioned compact toric surfaces affect their wve2c-graphs.

(iv) Since $q=p+1$ and gcd$(p+1,p-1)=$ gcd$(p+1,2)\in\{1,2\},$ formula (4.2) shows that the index $\ell$ of $X_{Q}\cong X_{Q_{p}^{\left[k\right]}}$ equals $\frac{p+1}{2}$ whenever $p$ is odd and $p+1$ whenever $p$ is even. This bears out our assertion about $\ell$. $\square$

Remark 5.6**.**

Among the LDP-polygons $Q_{p}^{\left[k\right]},$ only $Q_{1}^{\left[1\right]},Q_{1}^{\left[2\right]},Q_{1}^{\left[3\right]}$ are reflexive (with index $\ell=1$ and a unique Gorenstein singularity).

6. Defining equations

Let $Q$ be an arbitrary LDP-polygon. Since the Cartier divisor $-\ell K_{X_{Q}}$ on $X_{Q}$ is very ample, setting

[TABLE]

the complete linear system $\left|-\ell K_{X_{Q}}\right|$ induces the closed embedding $\Phi_{\left|-\ell K_{X_{Q}}\right|}$,

[TABLE]

with

[TABLE]

where $\chi^{(i,j)}:\mathbb{T}\rightarrow\mathbb{C}^{\ast}$ is the character associated to the lattice point $(i,j)$ (with $\mathbb{T}$ denoting the algebraic torus Hom${}_{\mathbb{Z}}(\mathbb{Z}^{2},\mathbb{C}^{\ast})$), for all $(i,j)\in(\ell\mathring{Q})\cap\mathbb{Z}^{2}.$ The image $\Phi_{\left|-\ell K_{X_{Q}}\right|}(X_{Q})$ of $X_{Q}$ under $\Phi_{\left|-\ell K_{X_{Q}}\right|}$ is the Zariski closure of Im$(\Phi_{\left|-\ell K_{X_{Q}}\right|}\circ\iota)$ in $\mathbb{P}_{\mathbb{C}}^{\delta_{Q}}$ and can be viewed as the projective variety Proj$(S_{\ell\mathring{Q}}),$ where

[TABLE]

$($with $C(\ell\mathring{Q}):=\{(\lambda y_{1},\lambda y_{2},\lambda)\left|\lambda\in\mathbb{R}_{\geq 0}\text{ and }(y_{1},y_{2})\in\ell\mathring{Q}\right.\})$ is the semigroup algebra which is naturally graded by setting deg$(\chi^{(i,j)}s^{\kappa}):=\kappa.$ (For a detailed exposition see [11, Theorem 2.3.1, p. 75; Proposition 5.4.7, pp. 237-238; Theorem 5.4.8, pp. 239-240, and Theorem 7.1.13, pp. 325-326].) Equivalently, it can be viewed as the zero set $\mathbb{V}(I_{\mathcal{A}_{Q}})\subset\mathbb{P}_{\mathbb{C}}^{\delta_{Q}}$ of the homogeneous ideal $I_{\mathcal{A}_{Q}}:=$ Ker$(\pi_{Q}),$ where

[TABLE]

and $\pi_{Q}$ is the $\mathbb{C}$-algebra homomorphism

[TABLE]

Furthermore, the projective degree $d_{Q}:=$ deg$(\mathbb{V}(I_{\mathcal{A}_{Q}}))$ of $\mathbb{V}(I_{\mathcal{A}_{Q}})$ (i.e., the double of the leading coefficient of the Hilbert polynomial of the homogeneous coordinate ring $\mathbb{C}[...:z_{(i,j)}:....]_{(i,j)\in(\ell\mathring{Q})\cap\mathbb{Z}^{2}}/I_{\mathcal{A}_{Q}}$) equals

[TABLE]

(See Sturmfels [40, Theorem 4.16, pp. 36-37, and p. 131] and [11, Proposition 9.4.3, pp. 432-433].)

Theorem 6.1** (Koelman [29]).**

If $\sharp(\partial(\ell\mathring{Q})\cap\mathbb{Z}^{2})\geq 4,$ then $I_{\mathcal{A}_{Q}}$ is generated by all possible quadratic binomials, i.e.,

[TABLE]

Corollary 6.2** (Castryck & Cools [9, §2]).**

If $\sharp(\partial(\ell\mathring{Q})\cap\mathbb{Z}^{2})\geq 4,$ and if we denote by $\beta_{Q}$ the cardinality of any minimal system of quadrics generating the ideal $I_{\mathcal{A}_{Q}},$ then

[TABLE]

Proof.

If HP${}_{2}(\mathbb{P}_{\mathbb{C}}^{\delta_{Q}}):=\left\{\text{homogeneous polynomials (in }\delta_{Q}+1\text{ variables) of degree }2\right\},$ then the $\mathbb{C}$-vector space homomorphism

[TABLE]

has as kernel Ker$(f)$ the $\mathbb{C}$-vector space of homogeneous polynomials of degree $2$ which belong to $I_{\mathcal{A}_{Q}}$ and as image Im$(f)$ the linear span of $\left.\{x^{i}y^{j}\right|(i,j)\in 2(\ell\mathring{Q})\cap\mathbb{Z}^{2}\}$ (because every lattice point in $2(\ell\mathring{Q})$ is the sum of two lattice points of $\ell\mathring{Q},$ cf. [11, Theorem 2.2.12, pp. 68-69]). Taking into account Koelman’s Theorem 6.1, [40, Lemma 4.1, p. 31], and the fact that $\mathbb{V}(I_{\mathcal{A}_{Q}})$ is not contained in any hyperplane of $\mathbb{P}_{\mathbb{C}}^{\delta_{Q}},$ the equality

[TABLE]

gives (6.2).

$\blacktriangleright$ Back to toric log del Pezzos with one singularity. Now let $Q$ be an LDP-polygon such that $X_{Q}$ has exactly one singularity. According to Theorem 1.4, there exist $p\in\mathbb{Z}_{>0}$ and $k\in\{1,2,3\},$ such that $X_{Q}\cong X_{Q_{p}^{\left[k\right]}}$ with index $\ell=\frac{p+1}{2}$ for $p$ odd and $\ell=p+1$ for $p$ even. For this reason, to apply Corollary 6.2 and to prove Theorem 1.5 we shall take a closer look at the dilated polars $\ell\mathring{Q}_{p}^{\left[k\right]}$ of the polygons $Q_{p}^{\left[k\right]}$ defined in (1.1).

Lemma 6.3**.**

The vertex sets of the polygons $\ell\mathring{Q}_{p}^{\left[k\right]},$ $k\in\{1,2,3\},$ are the following:**

[TABLE]

Proof.

Since $Q_{p}^{\left[1\right]}=$ conv$(\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\})$ with $\mathbf{v}_{1}=\tbinom{1}{-1},$ $\mathbf{v}_{2}=\tbinom{p}{1},\mathbf{v}_{3}=\tbinom{-1}{0},$ and

[TABLE]

with $l_{\text{conv}(\{\mathbf{v}_{1},\mathbf{v}_{2}\})}=\ell,$ $l_{\text{conv}(\{\mathbf{v}_{2},\mathbf{v}_{3}\})}=l_{\text{conv}(\{\mathbf{v}_{3},\mathbf{v}_{1}\})}=1,$ (4.3) gives

[TABLE]

Analogously, we conclude that

[TABLE]

After multiplication with the index $\ell$ we get $\mathcal{V}(\ell\mathring{Q}_{p}^{\left[k\right]}),$ $k\in\{1,2,3\}.$

Lemma 6.4**.**

The number of lattice points on $\partial(\ell\mathring{Q}_{p}^{\left[k\right]})$ is given in the table:**

[TABLE]

Proof.

Since the number of lattice points lying on the boundary of a lattice-polygon (w.r.t. $\mathbb{Z}^{2}$) is computed by the sum of the greatest common divisors of the differences of the vertex-coordinates of its edges, the above table is produced directly by using Lemma 6.3.

Remark 6.5**.**

Since $\sharp(\partial(\ell\mathring{Q}_{p}^{\left[k\right]})\cap\mathbb{Z}^{2})\geq 6$ for all $p\in\mathbb{Z}_{>0}$ and all $k\in\{1,2,3\},$ Theorem 6.1 and Corollary 6.2 can be applied for the LDP-polygons $Q_{p}^{\left[k\right]}.$

Lemma 6.6**.**

The projective degree $d_{Q_{p}^{\left[k\right]}}$ of $\mathbb{V}(I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})$ is given in the table:**

[TABLE]

Proof.

To determine the area of $\ell\mathring{Q}_{p}^{\left[k\right]}$ one may work with its vertex set given in Lemma 6.3. Alternatively, using [38, Proposition 2.10, p. 79] and formula (3.4) for $X_{Q_{p}^{\left[k\right]}}$ we deduce that

[TABLE]

and we read off $d_{Q_{p}^{\left[k\right]}}$ easier via (6.1) which gives $d_{Q_{p}^{\left[k\right]}}=\ell^{2}K_{X_{Q_{p}^{\left[k\right]}}}^{2}$.

Lemma 6.7**.**

The dimension $\delta_{Q_{p}^{\left[k\right]}}$ of the projective space in which $\mathbb{V}(I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})$ is embedded equals

[TABLE]

Proof.

(6.3) is immediate consequence of Pick’s formula.

Lemma 6.8**.**

The number $\beta_{Q_{p}^{\left[k\right]}}$ (of the elements of any minimal generating system of $I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})$ is given by the formula:

[TABLE]

Proof.

By the main properties of Ehrhart polynomial of the lattice polygon $\ell\mathring{Q}_{p}^{\left[k\right]}$ (cf. [11, Example 9.4.4, p. 433]) we obtain

[TABLE]

Hence, (6.4) follows from (6.2) and (6.1).

Hyperplanes $\mathcal{H}\subset\mathbb{P}_{\mathbb{C}}^{\delta_{Q_{p}^{\left[k\right]}}}$ give curves $\mathbb{V}(I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})\cap\mathcal{H}$ which are linearly equivalent to $-\ell K_{X_{Q_{p}^{\left[k\right]}}}.$ For generic $\mathcal{H}$’s the intersection $\mathcal{C}_{Q_{p}^{\left[k\right]}}:=\mathbb{V}(I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})\cap\mathcal{H}$ is (by Bertini’s Theorem) a smooth connected curve in the smooth locus of $\mathbb{V}(I_{\mathcal{A}_{Q_{p}^{\left[k\right]}}})\cong X_{Q_{p}^{\left[k\right]}}.$ The genus of $\mathcal{C}_{Q_{p}^{\left[k\right]}}$ is called the* sectional genus* $g_{Q_{p}^{\left[k\right]}}\emph{\vskip 6.0pt plus 2.0pt minus 2.0pt}$ of $X_{Q_{p}^{\left[k\right]}}.$

Lemma 6.9**.**

The sectional genus of $X_{Q_{p}^{\left[k\right]}}$ is

[TABLE]

Proof.

(6.5) follows from the fact that $g_{Q_{p}^{\left[k\right]}}=\sharp($int$(\ell\mathring{Q}_{p}^{\left[k\right]})\cap\mathbb{Z}^{2}).$ (See [11, Proposition 10.5.8, p. 509].)

Proof of Theorem 1.5: The number $\sharp(\partial(\ell\mathring{Q}_{p}^{\left[k\right]})\cap\mathbb{Z}^{2})$ and the projective degree $d_{Q_{p}^{\left[k\right]}}$ are known from Lemmas 6.4 and 6.6, respectively, while $\delta_{Q_{p}^{\left[k\right]}}$ is computed via (6.3), leading to Table (1.2), and consequently to Table (1.3) by making use of formula (6.4). Finally, one obtains Table (1.4) by means of the equality (6.5). $\square$

Note 6.10**.**

For a Magma code for the computation of a minimal generating system of the ideal defining the projective toric surface associated to an arbitrary lattice polygon, see [10]. In our particular case (in which we deal only with quadrics) it is enough to collect all vectorial relations $(i_{1},j_{1})+(i_{2},j_{2})=(i_{1}^{\prime},j_{1}^{\prime})+(i_{2}^{\prime},j_{2}^{\prime}),$ and to determine a $\mathbb{C}$-linearly independent subset of the set of the corresponding quadratic binomials $z_{(i_{1},j_{1})}z_{(i_{2},j_{2})}-z_{(i_{1}^{\prime},j_{1}^{\prime})}z_{(i_{2}^{\prime},j_{2}^{\prime})}$ by simply applying Gaussian elimination. For a short routine (written in Python) see [14].

Examples 6.11**.**

(i) The ideal $I_{\mathcal{A}_{Q_{1}^{\left[2\right]}}}$ (with $\mathbb{V}(I_{\mathcal{A}_{Q_{1}^{\left[2\right]}}})\subset\mathbb{P}_{\mathbb{C}}^{7}$) is minimally generated by the following $14$ quadrics:

[TABLE]

(ii) Correspondingly, the $9$ quadrics

[TABLE]

build a minimal set of generators of the ideal $I_{\mathcal{A}_{Q_{1}^{\left[3\right]}}},$ and $\mathbb{V}(I_{\mathcal{A}_{Q_{1}^{\left[3\right]}}})\subset\mathbb{P}_{\mathbb{C}}^{6}.$ ($X_{Q_{1}^{\left[3\right]}}$ is obtained by blowing up $X_{Q_{1}^{\left[2\right]}}$ at one non-singular point, cf. Figure 5.)

(iii) The next coming example, namely that one created by the LDP-polygon $Q_{3}^{\left[3\right]}$ (cf. Figure 6), in which $2\mathring{Q}_{3}^{\left[3\right]}\cap\mathbb{Z}^{2}$ consists of $22$ lattice points and $\mathbb{V}(I_{\mathcal{A}_{Q_{3}^{\left[3\right]}}})\subset\mathbb{P}_{\mathbb{C}}^{21},$ is much more complicated. Using [14] we see that $I_{\mathcal{A}_{Q_{3}^{\left[3\right]}}}$ is minimally generated by the following $182$ quadrics:

[TABLE]

[TABLE]