Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua

When F-theory is compactified on a compact Calabi-Yau threefold $X$ that is elliptically fibered, one gets a 6D theory of gravity with $\mathcal{N}=1$ supersymmetry. This theory may also contain tensor, vector, and scalar (hypermultiplet) fields.

The elliptic fibration $\pi:X\rightarrow S$ must have a global section; a Weierstrass model of any such fibration exists Nakayama and takes the form

If we denote the canonical class of the base by $K_{S}$, then the functions $f$, $g$ and $\Delta$ are sections of line bundles ${\mathcal{O}}(-4K_{S})$, ${\mathcal{O}}(-6K_{S})$ and ${\mathcal{O}}(-12K_{S})$ respectively.

The fiber is singular along loci in the base manifold $S$ where the discriminant

vanishes. The Kodaira-Tate classification characterizes codimension-one singularities where the discriminant $\Delta$ vanishes Kodaira ; Tate . When the functions $f$, $g$ and $\Delta$ vanish to specific orders on a divisor $D$, the singularity is associated with the Dynkin diagram of a certain non-Abelian gauge algebra that describes a gauge field in the corresponding physical 6D supergravity theory Morrison-Vafa-I ; Morrison-Vafa-II ; Bershadsky-all ; Morrison-sn ; Grassi-Morrison-2 . The correspondence between vanishing orders of $f,g$ and nonabelian symmetry algebras is listed in Table 1. Codimension two singularities give rise to matter fields in the 6D theory, though this correspondence is not yet completely classified. Note that if $(f,g)$ vanish to order $(4,6)$ or higher at a codimension-one locus on the base, the singular Weierstrass model (1) cannot be resolved into a smooth Calabi-Yau threefold. Then the F-theory compactification does not have a supersymmetric M-theory dual, hence it will not give any supergravity vacua. These points are also at infinite distance in the Weierstrass moduli space. If $(f,g)$ vanishes to order $(4,6)$ or higher at a codimension-two locus (points) on the base, then such points need to be blown up on the base in the resolution process to give a smooth Calabi-Yau threefold.

Rational curves of negative self-intersection on the base surface $S$ can enforce minimal degrees of vanishing of $f$, $g$, $\Delta$ over those curves, for a generic elliptic fibration. When this happens, the physical theory has some minimal content of gauge fields and matter fields, regardless of the specific fibration one chooses in the Weierstrass moduli space. The configurations of curves of negative self-intersection that give rise to gauge and matter fields in this way were classified in Classbasis , and dubbed “non-Higgsable clusters” (NHCs), since the 6D gauge groups that arise in this way cannot be broken in a fashion that preserves supersymmetry by giving vacuum expectation values to (i.e., “Higgsing”) the charged matter fields. We have listed the configurations of possible NHCs in Table 2. While the gauge groups produced by non-Higgsable clusters cannot be broken in any supersymmetric vacuum without changing the base $S$, the coefficients of the Weierstrass model, associated with neutral scalar fields in the 6D supergravity theory, may be tuned to enhance the gauge contents of the theory. It was shown, for example, that in this way all elliptically fibered Calabi-Yau threefolds with $h^{2,1}(X)\geq 350$ (and indeed all known Calabi-Yau threefolds exceeding this Hodge number bound) can be constructed through tuning Weierstrass models over simple complex base surfaces JohnsonTaylor .

The Hodge numbers of the generic (i.e., untuned) elliptic Calabi-Yau threefold $X$ associated to a given base $S$ can be easily computed from the non-Higgsable cluster content of the base WT-Hodge using the relations

Here, as above, $T=h^{1,1}(S)-1$ is the number of tensor multiplets. The quantity $R$ is the total rank of (nonabelian + abelian) gauge groups. $V$ and $H_{\mathrm{charged}}$ are the numbers of 6D vector supermultiplets and charged matter hypermultiplets respectively Morrison-Vafa-I ; Morrison-Vafa-II . $V$, $H_{\mathrm{charged}}$, and the contribution of nonabelian gauge groups to $R$ can be directly computed by summing up the contributions of all NHCs, using Table 2, when the intersection structure of curves on the base is known. In principle, there can also be an abelian contribution to $R$, even when $X$ is the generic elliptic CY threefold over the base $S$. While this does not occur for toric base surfaces $S$, it was found in semi-toric that there are a small number of base surfaces with a single $\mathbb{C}^{*}$ structure (“semi-toric” bases) that give rise to theories with a non-Higgsable abelian gauge group structure. The corresponding Calabi-Yau threefolds are elliptic fibrations with multiple independent sections, corresponding to a higher rank Mordell-Weil group. It is shown in Morrison-Park-Taylor that all these threefolds are related and can be described through a generalization of the Schoen construction Schoen . In general, computing the rank of the Mordell-Weil group for a given compactification space is a difficult mathematical problem, which we do not attempt to address here. For the specific classes of non-toric bases that we enumerate here, there are no abelian contributions to $R$ for generic elliptic fibrations. For a comprehensive analysis of bases giving elliptic Calabi-Yau threefolds with large $h^{1,1}$ and small $h^{2,1}$, however, this possibility would need to be considered further.

As mentioned above, the base $S$ of any elliptically fibered Calabi-Yau threefold $X$ can be constructed by blowing up a finite number of points on $\mathbb{P}^{2}$, a Hirzebruch surface $\mathbb{F}_{n}(0\leq n\leq 12)$ or the Enriques surface. The F-theory models in the case of the Enriques surface are relatively trivial, and do not allow any gauge field or matter content, since $-K_{S}$ vanishes (up to torsion). In this paper we only consider $\mathbb{P}^{2}$ and $\mathbb{F}_{n}(0\leq n\leq 12)$ Classbasis . In Kodaira’s classification of complex surfaces (not to be confused with his classification of codimension one singularities in elliptic fibers mentioned above), these are all rational surfaces (birational to $\mathbb{P}^{2}$), with Kodaira dimension $\kappa(S)=-\infty$ bhpv . The condition that $-4K_{S},-6K_{S},-12K_{S}$ have global sections means that the anticanonical class $-K_{S}$ is in the cone of effective divisors. Stated more technically, the complex surfaces we study in this paper are anticanonical rational surfaces Looijenga ; Harbourne1997 , or those where the anticanonical class is an effective $\mathbb{Q}$-divisor. We focus in this paper only on smooth base surfaces $S$.

The general theory of complex algebraic surfaces can be found in Griffith-Harris ; bhpv . We review here a few basic definitions and ideas that will be helpful in systematically generating base surfaces.

A (Weil) divisor $D$ is a formal linear combination with integer coefficients of irreducible algebraic curves on the algebraic surface $S$. If the coefficients in the linear combination are all non-negative, then the divisor is effective, $D\geq 0$. A $\mathbb{Q}$-divisor is defined similarly to a Weil divisor, but with coefficients in $\mathbb{Q}$ (Note that Weil divisors and Cartier divisors coincide on smooth varieties, which is the only situation that we consider here). The divisor class $[D]$ is the homology class of $D$. The set of divisor classes modulo linear equivalence on $S$ forms an additive group, which is homomorphic to the Picard group Pic$(S)$ of line bundles on the surface (which form a group under the tensor product operation). For effective divisors, the divisors in the same divisor class as a given divisor $D$ are said to be linearly equivalent, and form a linear system, denoted $|D|$.

There is a bilinear operation $\cdot:$Pic$(S)\times$Pic$(S)\rightarrow\mathbb{Z}$, which corresponds to the intersection number of two divisor classes. For two distinct divisor classes $[D_{1}],[D_{2}]$, we can pick out one specific representative $D_{1}\in|D_{1}|,D_{2}\in|D_{2}|$ from each class; the intersection number $[D_{1}]\cdot[D_{2}]$ then just coincides with the geometric number of intersection points (with multiplicity) between the two curves $D_{1}$ and $D_{2}$. The self-intersection of a divisor class $[D]$ is $[D]\cdot[D]$. Note that the self-intersection of a curve can be negative, but only when the curve is rigid and cannot be deformed; otherwise a small deformation would give two distinct curves that would intersect at a finite positive number of points. (because of the complex structure, all intersections between complex curves in a surface have the same orientation, and all contribute positively to intersection numbers.)

For $\mathbb{P}^{2}$, the inequivalent divisor classes are those of the form $n[H]$, where $n\in\mathbb{Z}$ and $[H]$ is the divisor class of the hyperplane in $\mathbb{P}^{2}$; the hyperplane $[H]$ is just a line $\mathbb{P}^{1}\subset\mathbb{P}^{2}$ given by a linear equation, for example as described by the algebraic equation $z=0$ in homogeneous coordinates $[x:y:z]$ on $\mathbb{P}^{2}$. The linear system $|nH|$ corresponds to the space of all degree-$n$ curves on $\mathbb{P}^{2}$.

The mathematical procedure of blowing up a point $p$ on a smooth surface $S$ consists of replacing the point by a copy of $\mathbb{P}^{1}$, called the exceptional curve, where the points on the $\mathbb{P}^{1}$ correspond to all (complex) directions in which lines on the original surface can pass through the original point. After a blow-up process, a new effective divisor class arises on the resulting blown up surface $S^{\prime}$, which is the class of the exceptional curve $[E]$. This divisor class has self-intersection $-1$; the linear system of the exceptional curve only contains one curve, hence it is generally just called the exceptional curve. There is a birational map $\pi:S^{\prime}\rightarrow S$, which is one-to-one and onto everywhere except on the exceptional curve, where $\pi:E\rightarrow p$ and $p$ is the point being blown up. If there is some representative curve $D$ of a certain effective divisor class $[D]$ that passes through the point $p$, with multiplicity $m$, then after the blow-up process, a new divisor class

appears as an effective divisor class on $S^{\prime}$. This is called the proper transform of $D$. Here we have identified $[D]$ on $S^{\prime}$ with $\pi^{-1}([D])$.

For a given surface $S$ generated by blowing up $\mathbb{P}^{2}$ consecutively $r$ times, a convenient basis of Pic$(S)$ consists of the divisor class of the hyperplane $[H]$ on the original $\mathbb{P}^{2}$ and the exceptional divisors $[E_{1}],[E_{2}],\dots,[E_{r}]$. The intersection matrix on this basis is given by

Hence the Picard rank is rk(Pic$(S))=r+1$.

More generally, the Hodge index theorem states that for any surface with rk(Pic$(S)$) $=r+1$, the signature of the intersection matrix is $(1,r)$. When $r>1$, there is always a basis in which the intersection product takes the form (6); for $r=1$, there are surfaces where the intersection form has the structure of the matrix

The canonical divisor class of $\mathbb{P}^{2}$ is $[K_{S}]=-3[H]$, and after $r$ blow-ups is always in the form of

More generally, the canonical divisor class can always be taken to have this form in a basis with intersection form (6). In general, if an irreducible divisor class on a surface formed from $r$ blow-ups on $\mathbb{P}^{2}$ can be decomposed into a sum of the hyperplane and other divisor classes as $[D]=n[H]+\dots$, then it is the divisor class of some degree $n$ curve on the original $\mathbb{P}^{2}$.

There is a useful genus formula that follows from the adjunction formula:

where $g$ is the genus of the curve $C$. If $g=0$, then $C$ is a rational curve. If $g=1$, then $C$ is an elliptic curve. Note that the anticanonical divisor class $[-K_{S}]$ is always elliptic.

The dimension of the linear system $|D|$ is related to the dimension of the cohomology class of sheaves $\mathcal{O}([D])$:

For rational curves $D$, we have the following result from the Riemann-Roch theorem Kollar ; CaporasoHarris ; CaporasoHarris2

If $\mathrm{dim}(|D|)>0$, then there exists a curve $D\in|D|$ that passes through $\mathrm{dim}(|D|)$ general points. This implies furthermore that if $[D]\cdot[C]>0$ for another divisor class $[C]$, then for every generic point $p$ on $C\in|C|$, there exists a curve $D\in|D|$ that passes through it. This is the case whenever the class of the divisor $D$ has non-negative self-intersection. When the class $[D]$ has negative self-intersection, then $\mathrm{dim}(|D|)\leq 0$, which means that the curve in $|D|$ cannot be deformed. In this case, if $[D]$ is irreducible, then it is really a single “fixed” curve.

For general (arbitrary genus) curves, the value of $h^{0}(S,\mathcal{O}([D]))$ satisfies $h^{0}(S,{\mathcal{O}}(D))\geq([D]^{2}-[K_{S}]\cdot[D])/2+1$ Griffith-Harris .

In general we have the following lemma:

More generally, this lemma also guarantees that curves of sufficiently positive self-intersection can pass through points with arbitrary multiplicity. When $([D]^{2}-[K_{S}]\cdot[D])/2\geq 3$, for example, there is a representative curve $D$ that passes through any point $p$ in $S$ with multiplicity 2 (which requires two additional conditions, corresponding to the vanishing of the two first derivatives at $p$). In general, for a curve to exist that passes through any point with multiplicity $m$, we must have $([D]^{2}-[K_{S}]\cdot[D])/2\geq m(m+1)/2$.

Note that the above statements are completely consistent with the intuition of planar geometry. For example, consider the curves of self-intersection $(-1)$ on a surface formed by blowing up $\mathbb{P}^{2}$ $r$ times. The divisor classes of degree 1 rational curves with self-intersection $(-1)$ can be written as $[H]-[E_{i}]-[E_{j}]$, corresponding to the set of lines that pass through the pairs of points $p_{i}$ and $p_{j}$ on the original $\mathbb{P}^{2}$. There is a unique line that passes through two fixed points in the plane, hence the resulting $(-1)$-curve is fixed. Similarly, the divisor class of degree 2 rational curves with self-intersection $(-1)$ can be written as $2[H]-[E_{i}]-[E_{j}]-[E_{k}]-[E_{l}]-[E_{m}]$, corresponding to the set of conics that pass through five fixed points. We know this divisor is also rigid, since there is a unique conic passing through five fixed points. If we consider, however, the divisor class of degree 1 rational curves with self-intersection 0, $[H]-[E_{i}]$, these correspond to the set of lines that pass through a particular point $p_{i}$, and are free to move. In fact, such lines can pass through through every point on $\mathbb{P}^{2}$.

We have focused so far on blow-ups of $\mathbb{P}^{2}$, but the above properties of divisors can also be applied to the Hirzebruch surfaces $\mathbb{F}_{n}$ (Note that $\mathbb{F}_{1}$ is the surface that results from blowing up a single point on $\mathbb{P}^{2}$). In the language of toric geometry, the irreducible toric divisors form a cyclic diagram (see Figure 1). For the cases with odd $n=2k-1$, the relevant divisor classes can be written in the basis (6) using the following linear combinations of the two generators of Pic$(\mathbb{F}_{2k-1})$:

The canonical class is, for all $k$,

The blowing-up of $\mathbb{F}_{2k-1}$ at $r-1$ points has the same Picard groups and intersection matrices for each $k$, and matches with that of $\mathbb{P}^{2}$ blown up at $r$ points. Also, the canonical class is always again in the form $K=-3[H]+\sum_{i=1}^{r}[E_{i}]$. Generally speaking, therefore, although the geometric intuition comes most easily from $\mathbb{P}^{2}$, the structure of divisors and intersection structure is essentially the same on other rational surfaces arising as blowups of Hirzebruch surfaces.

For even $n$, we can relate any blow-up of the Hirzebruch surface $\mathbb{F}_{n}$ to the cases with odd $n$ via the blow-up chains shown in Figure 2. Thus, every surface that results from blowing up $\mathbb{F}_{2k}$ can be generated by blowing up $\mathbb{F}_{2k-1}$ or $\mathbb{F}_{2k+1}$. Hence, to enumerate all possible base surfaces with $r>1$ that support elliptic Calabi-Yau threefolds we only need to take $\mathbb{F}_{n}$ with odd $n$ as starting points for the blowing up process.

The set of effective divisor classes on $S$ forms a cone, called the effective cone Eff$(S)$. The dual of the effective cone is the set of divisor classes that intersect non-negatively with $[C]\in$Eff$(S)$; this is called the nef cone Nef$(S)$:

Elements of the nef cone are called nef divisors (classes). In general, an element in the effective cone need not be nef; for example, any effective divisor of negative self-intersection is not nef. Conversely, however (see e.g. Corollary II.3 in Harbourne1997 ), any nef divisor class on a smooth rational surface is effective.

The effective cone is the primary distinguishing characteristic of different rational surfaces. For example, the Hirzebruch surfaces $\mathbb{F}_{n}$ with $n$ odd all have the same intersection form as discussed above. The cone of effective divisors, however, is generated by $[S_{\infty}]$ and $[F]$, and is different for each $n$.

Our primary tool in this paper for studying base surfaces $S$ will be the combinatorial structure of the effective cone. The following result, which is Proposition 1.1 in Cox-rings , will be useful

For the Hirzebruch surfaces, of Picard rank 2, it follows directly from the explicit description above that the effective cone is generated by rational divisor classes with non-positive self-intersection (i.e., $[S_{\infty}],[F]$, where $[F]\cdot[F]=0,[S_{\infty}]\cdot[S_{\infty}]=-m$ for $\mathbb{F}_{m}$). After one blow-up, Lemma 2 applies. Blowing up $\mathbb{F}_{m}$ ($m$ odd), an effective class $[F]-[E_{2}]=[H]-[E_{1}]-[E_{2}]$ with $(-1)$ self-intersection always appears. Then the effective class with 0 self-intersection can be written as a non-negative linear combination of $[H]-[E_{1}]-[E_{2}]$ and exceptional divisors. This property generally holds for surfaces with higher Picard rank.

We discuss these statements further in section 4.

Note that curves of negative self-intersection on a base surface that supports an elliptic Calabi-Yau must always be rational. From (9), an irreducible curve of negative self-intersection with $g\geq 1$ satisfies $-K\cdot C\leq C\cdot C<0$. From this it follows that $-nK$ contains $C$ at least $n$ times. This means that $f,g$ must vanish to at least orders $4,6$ over $C$ so we cannot have such a curve in a base that supports an elliptic Calabi-Yau threefold.

To close this section, we introduce some terminology that we will use frequently in the remainder of the paper.

• •

  In general we will not distinguish between “divisor class”, “divisor” and “curves”. Thus, for example, the divisor class $[D]$ is generally written as $D$. When we talk about “$n$-curves”, this refers to a divisor class with self-intersection $n$. We often use the term “negative curve” to refer to a curve of negative self-intersection.

• •

  Following Lemma 2 and the subsequent discussion, the negative curves that appear in the surfaces in which we are interested in are always rational. Furthermore, in the cases where Lemma 2 applies, we define Neg$(S)$ as the set of irreducible negative rational curves that generate Eff$(S)$. A general requirement for Neg$(S)$ is:

  For $C,D\in\mathrm{Neg}(S)$, if $C\neq D$, then $C\cdot D\geq 0$. This is just the statement that two different irreducible curves cannot intersect each other negatively.

• •

  We further define the set of irreducible rational curves $C$ with self intersection number less than -1 to be Sing$(S)$. These include the non-Higgsable clusters described in Table 2, as well as configurations of $(-2)$-curves whose intersection matrices are exactly the Cartan matrices for ADE Lie algebras.

• •

  When discussing toric bases, we indicate the cycle of self-intersection numbers by e.g. $((n,0,-n,0))$.

• •

  We will use the vector representation $(a_{0},a_{1},\dots,a_{r})$ for the divisor class $D=a_{0}H+\sum_{i=1}^{r}a_{r}E_{r}$. This applies to surfaces that arise from blowing up any $\mathbb{F}_{2k-1}$ any number of times.

• •

  We define the set of genus $g$, self-intersection $k\geq 0$ divisors $C$ that intersect non-negatively with every curve $D\in\mathrm{Neg}(S)$ by $\mathfrak{C}_{g,k\geq 0}(S)\subset\mathrm{Nef}(S)$. They are all effective curves, as we mentioned before. All such curves can be described as integer vectors as described in the previous point.

  Finally we define $\mathfrak{C}_{k}(S)=\bigcup_{g}\mathfrak{C}_{g,k}(S)$.

• •

  Using the terminology in Derenthal ; DerenthalThesis , for two surfaces $S$ and $R$, if there is a bijection between Neg$(S)$ and Neg$(R)$ that preserves the intersection structure then Eff$(S)$ and Eff$(R)$ are isomorphic, and we say that $S$ and $R$ are “of the same type”. This equivalence relation is weaker than isomorphism, as demonstrated in DerenthalThesis . Nevertheless, two surfaces of the same type give the same minimal gauge algebra. The blow-up descendants of two surfaces of the same type may be different, as will be discussed in section 5.

• •

  We always denote the one-time blow-up of the surface $S$ by $S^{\prime}$

As mentioned before, when we blow up a point that lies with multiplicity $m$ on a representative of an effective divisor class $C$ and self-intersection $k$, there is a new effective divisor class $C^{\prime}=(C,-m)$, written in the vector representation. Using the Riemann-Roch formula (9), the following lemma holds:

This implies that blowing up a single point on a curve will not change its (arithmetic) genus, but blowing up an $m$-point on a curve will decrease the genus of the curve by $(m^{2}-m)/2$.

Then given a base $S$, the different ways of blowing it up can be classified as follows:

As a consistency requirement, the resulting rational curves after the blow up should intersect non-negatively with each other. This is equivalent to the following statement:

For $A\in\mathfrak{C}_{(m^{2}-m)/2,m^{2}-1}(S)$, $B\in\mathfrak{C}_{(n^{2}-n)/2,n^{2}-1}(S)$ that are blown up in this process, $A\cdot B\geq mn$ ($m,n>0$).

(2) One can choose to blow up a non-generic point, so that a set of curves $C_{0,i}\in\mathfrak{C}_{0,k<0}$ are blown up at a single point. The index $i$ here just labels different curves with same genus. In the simplest cases, the point blown up lies only on a single rational curve $C$ of negative self-intersection, in which case the transformed curve is $C^{\prime}=(C,-1)$. The point blown up may also lie at an intersection point between a pair of negative curves $C,D$, in which case both curves are transformed and $C^{\prime}\cdot D^{\prime}=0$.

(3) There are also situations where a set of curves $C_{1,i}\in\mathfrak{C}_{1,k<3}$ are blown up at a double point, $C_{2,i}\in\mathfrak{C}_{3,k<8}$ are blown at a triple point, and so on. This will produce new negative rational curves of self-intersection $<-1$ that must be included in Neg$(S^{\prime})$ We define the blow-up process to be a “special blow-up” when one or more (-2) or lower curves are generated by blowing up positive curves at points with multiplicity higher than 1.

(4) In some cases it may be possible to choose a non-generic point as in (2) that lies at the intersection of more than two negative curves.

Cases (1) and (2) can be handled in a systematic fashion using the combinatorial data of the effective cone. Case (3) may or may not give us new bases. In the specific regimes that are thoroughly studied in this paper, special blow-ups are observed not to occur. Case (4) is discussed further in section 6, and also does not occur in the regimes that we study explicitly.

Just as for blowups producing $-1$ curves, we require that for intersecting curves $C_{(m^{2}-m)/2,i}\in\mathfrak{C}_{(m^{2}-m)/2,k<m^{2}-1}$, $C_{(n^{2}-n)/2,j}\in\mathfrak{C}_{(n^{2}-n)/2,k<n^{2}-1}$, that are blown up at points of multiplicity $m,n$ respectively, we must have $C_{(m^{2}-m)/2,i}\cdot C_{(n^{2}-n)/2,j}\geq mn$. Otherwise after the blow up, Neg$(S^{\prime})$ will contain two different elements that intersect each other negatively.

From the analysis above, all the curves in the set Sing$(S^{\prime})$ can be generated by blowing up non-generic points. However, there is also in general a new set of (-1)-curves in Neg$(S^{\prime})$, and it is not clear whether the algorithm generating (-1)-curves described above is finite or not. In principle we need to look at all the sets $\mathfrak{C}_{(m^{2}-m)/2,m^{2}-1}$, but this is impossible to do. Now we prove a powerful proposition, which contains the methodology to generate all the $(-1)$-curves with a finite amount of work:

Actually the two equations are nothing but the defining equations of self-intersection and genus.

Actually, it is known that Proposition 1 holds for generalized del Pezzo surfaces Derenthal ; DerenthalThesis . We just needed to confirm that it holds for the bases that we are interested in.

Now the question is how to generate the solutions to the Diophantine equations using a finite algorithm. In fact, all the solutions can be generated by a series of “$q$-operations” acting on curves, which are defined as follows Nagata :

For a curve in the form $(a,b_{1},b_{2},\dots,b_{r})$, one picks three numbers $i_{1},i_{2},i_{3}$ out of $\{1,\ldots,r\}$, then performs the following transformation:

It can be explicitly checked that the two quantities $a^{2}-\sum_{i=1}^{r}b_{i}^{2}$ and $3a+\sum_{i=1}^{r}b_{i}$ do not change under this transformation. In fact, this operation is the Cremona transformation with center $p_{i_{1}},p_{i_{2}},p_{i_{3}}$.

In practice, one starts from some low-degree curves, such as all the degree 0 and degree 1 rational (-1)-curves. Then one tries to perform all the $q$-operations on one curve so that the degree is increased ($d>0$ in (27) ). If the curve intersects negatively with some element in Sing$(S)$, then we cut this branch. This recursive algorithm is finite if Eff$(S)$ is finitely generated. We discuss more about the finiteness of curves in section 6.

There are some cases, however, where a high degree (-1)-curve exists but there are no valid degree 1 curves. If this happens, the set of (-1)-curves we get from the recursive algorithm above is not complete. This is equivalent to the statement that there is another (-1) curve that intersects non-negatively with all the curves we have found, but it is not in the solution set generated by carrying out the algorithm up to a fixed degree $d$. Such an additional curve would appear to be a negative curve in the nef cone, which is impossible. While the full set of (-1)-curves would in principle be produced by keeping all intermediate branches and continuing the algorithm to arbitrary $d$, without some further check on the completeness of the set of generators for the effective cone we would not have a means of terminating the algorithm. So for a consistency check we compute the dual cone of Neg$(S)$ generated by the method above, and check that the generators (extremal rays) of this dual cone are non-negative. If a generator is negative, we know it is a (-1)-curve in Neg$(S)$ that was not generated by the $q$-process. After adding all the (-1)-curves of this kind, we can get a complete set of curves in Neg$(S)$, with a dual cone that contains no negative curves.

Another more complicated issue is the “special blow-up” that was defined earlier, where one blows up a non-generic point on a curve with genus higher than 1 to get an element in Sing$(S^{\prime})$. This also appears to make the algorithm infinite, since in principle we need to examine the set of all positive curves. This issue appears to be difficult to handle in a completely general context, and we will discuss this issue in specific regimes.