Classifying Fano Complexity-One $T$-Varieties via Divisorial Polytopes
The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one -varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger’s classification of Gorenstein del Pezzo -surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions.
It is well-known that Fano toric varieties with at worst Gorenstein singularities correspond to so-called reflexive polytopes, see e.g. [CLS11, §8.3]. Furthermore, in any fixed dimension, there are only a finite number of reflexive polytopes up to equivalence [LZ91], and there is an algorithm for classifying them [KS97]. It follows that there are only finitely many Gorenstein Fano toric varieties in a fixed dimension, and they may be classified algorithmically.
A natural generalization of toric varieties are complexity-one -varieties: normal varieties equipped with the effective action of an algebraic torus whose generic orbit has codimension one in . Any toric variety may be considered as a complexity-one -variety by restricting the action of the big torus on to a codimension-one subtorus .
In a manner similar to the case of toric varieties, these varieties can also be encoded using quasi-combinatorial data, specifically, a generalization of polytopes. Throughout, suppose is a lattice.
A combinatorial divisorial polytope (CDP) with respect to the lattice consists of a full-dimensional lattice polytope , along with an -tuple (for some ) of piecewise-affine concave functions such that
For each , the graph of is a polyhedral complex with integral vertices;
For each , .
We call the base of . The dimension of is .
Attaching each of the functions to a point in the curve gives rise to a divisorial polytope on ; these correspond to rational polarized complexity-one -varieties [IS11].
Just as there is a special subclass of lattice polytopes corresponding to Gorenstein Fano toric varieties, there is a special subclass of CDPs corresponding to Fano complexity-one -varieties with at worst canonical Gorenstein singularities [IS17, Definition 3.3 and Theorem 3.5]. We call such CDPs Fano, and recall the details in Definition 2.3.
Considering the finiteness result for reflexive polytopes, one might ask if a similar result holds for Fano CDPs. To pose this question we define a natural notion of equivalence of CDPs in § 2.1. We say that a CDP is toric if it is equivalent to a CDP consisting of at most two functions . Our conjecture for Fano CDPs is the following:
In any fixed dimension , there are only finitely many equivalence classes of non-toric Fano CDPs.
Geometrically, this conjecture is equivalent to stating that there are only finitely many families of canonical Gorenstein Fano complexity-one -varieties in any given dimension. It should be noted that any Fano toric variety can be considered as a complexity-one -variety in infinitely many ways, which is why we exclude the toric case from the above conjecture.
1.1. Main results
Our first main result is the following:
Main Theorem 1 (Theorem 4.2).
Over any fixed base , there are at most finitely many equivalence classes of Fano CDPs.
While we do not present it as such, this result can be made effective. The proof of Main Theorem 1 suggests an algorithm to enumerate all equivalence classes of Fano CDPs over a fixed base.
Given Main Theorem 1, to prove Conjecture 1.2 it is sufficient to prove that only finitely many base polytopes occur for non-toric Fano CDPs in any given dimension. Equivalently, a bound on the volume of the bases which occur (or the number of interior lattice points [LZ91]) would also prove the conjecture.
We then apply the tools we develop in the proof of Main Theorem 1 to analyze low-dimensional cases:
Main Theorem 2 (Theorem 5.1).
There are exactly 34 equivalence classes of two-dimensional non-toric Fano CDPs.
Main Theorem 3 (Theorem 6.1).
Any three-dimensional Fano CDP is equivalent to one with at most eight functions.
Our analysis in these cases leads us to conjecture (Conjecture 6.4) that any -dimensional Fano CDP is equivalent to one with at most functions.
1.2. Geometric interpretations of the main results
Our results have an interesting geometric interpretation. Given a Fano complexity-one -variety with at worst canonical Gorenstein singularities, let be the resolution of the rational quotient map .
For a fixed polarized toric variety , there are finitely many families of canonical Gorenstein Fano complexity-one -varieties such that the general fiber of is isomorphic to , polarized with regards to the pullback of .
There are exactly 34 families of non-toric Gorenstein del Pezzo -surfaces .
For any canonical Gorenstein Fano complexity-one threefold, the resolved quotient map has at most non-integral fibers.
An alternate approach to the classification of Fano complexity-one -varieties is via a classification of their Cox rings. This has been employed by Huggenberger [Hug13, Hau13] to classify Gorenstein del Pezzo surfaces with a -action; our Main Theorem 2 and Corollary 1.4 recover this classification. A translation of this classification into the language of divisorial polytopes used here is found in [CS17].
This approach via Cox rings has also been employed in [HHS11, BHHN16, AF17] to classify higher-dimensional Fano complexity-one -varieties of Picard ranks one and two. These techniques should work more generally, as long as one has a bound on the Picard rank of the varieties in question. Interestingly, while our Main Theorem 3 does not bound the Picard rank of Fano complexity-one threefolds, it does limit the number of equations which may appear in the defining ideal of the Cox ring. We hope that by combining this bound with the Cox ring approach, one may obtain a complete classification of canonical Gorenstein Fano complexity-one threefolds.
1.3. Outline of the article
First we provide some background information on CDPs in §2, including a description of operations preserving equivalence, as well as establishing the relationship between lattice polytopes and toric CDPs. We recall the definition of a Fano CDP given by Ilten and Süß in [IS17], and use the notion of equivalence to make some key simplifying assumptions about the properties of Fano CDPs. In §3 we provide, for fixed base polytope , an upper bound on the number of functions in a Fano CDP. Main Theorem 1, stating the finiteness of the number of equivalence classes of Fano CDPs over any fixed base, is established in §4. In §5, we use the tools developed throughout this paper to provide a classification of all equivalence classes of two-dimensional Fano CDPs, proving Main Theorem 2. Finally, in §6, we prove Main Theorem 3.
2. Generalizing reflexive polytopes to Fano CDPs
In this section, we introduce the notion of equivalence of CDPs, discuss how CDPs can be viewed as a generalization of lattice polytopes, and introduce the Fano property of CDPs.
2.1. Equivalence of CDPs
Let be a -dimensional CDP with base and functions . We may perform any combination of the following actions or their inverses to obtain an equivalent CDP:
- Addition of the zero function.
If , then is equivalent to .
For , the CDP with base and functions
is equivalent to .
- Transformation of the base.
If is an invertible affine linear transformation of the lattice , the CDP with base and functions is equivalent to .
For with , the CDP with base and functions is equivalent to .
- Shearing Action.
For and with , the CDP with base and functions
is equivalent to .
In Figure 2.1, we illustrate the equivalence operations of shearing, translating, and transforming the base.
The equivalences are motivated geometrically as follows. Given a CDP , attaching points to each function gives rise to a divisorial polytope on ; these correspond to rational polarized complexity-one -varieties [IS11]. Any two equivalent CDPs will give rise to complexity-one -varieties which are equivariantly isomorphic, after appropriate choice of the points .
2.2. From polytopes to toric CDPs
Consider a polytope in , with vertices in the lattice . This gives rise to a CDP with two functions as follows. Let be the projection to and be the projection to . Set and define by
Then the base with the functions , is a CDP. This process is illustrated with an example in Figure 2. Conversely, this process can be inverted to obtain a lattice polytope from a CDP with two functions by reflecting one of the functions and taking the convex hull of the vertices of the graph of the CDP.
A CDP is toric if it is equivalent to a CDP with at most two functions. Since we may add a constant zero function to obtain an equivalent CDP, we have that toric CDPs are those equivalent to a CDP with exactly two functions. Thus, toric CDPs are exactly those which arise from a lattice polytope via the above construction.
If two toric CDPs are equivalent, then their corresponding polytopes are equivalent as lattice polytopes.
The converse of Remark 2.2 does not hold; that is, equivalent polytopes do not necessarily correspond to equivalent CDPs. Examples of this fact can be found by shearing a polytope in a direction other than the direction that we project along to obtain the polytope’s corresponding CDP. Such an example is given in Figure 2.2; the CDPs depicted there are inequivalent because their base polytopes are inequivalent. From the point of view of algebraic geometry, this is saying that a toric variety may be given the structure of a complexity-one -variety by restricting to a codimension-one subtorus in infinitely many different ways.
2.3. Fano Divisorial Polytopes
The Fano property of a CDP, as defined by Ilten and Süß in [IS17], generalizes the reflexive property of polytopes. Under the correspondence between divisorial polytopes and polarized rational complexity-one -varieties, it corresponds exactly to canonical Gorenstein Fano -varieties with anti-canonical polarization. We recall this property below in the context of CDPs.
For a polytope , we denote its interior by and its boundary by . For a function , we denote the graph of by . A facet of a lattice polytope in is in height one if there exists some such that for all .
A CDP is Fano if it is equivalent to a CDP with base polytope and functions for which there are integers such that
For all , , and each facet of is at height one;
For any facet of not at height one, on .
When each facet of is at height one, we say that is at height one.
The four properties of Definition 2.3 are preserved for equivalent CDPs, provided that any transformation of the base preserves the origin.
It is straightforward to check that under the construction producing a toric CDP from a lattice polytope in , the CDP is Fano if and only if its corresponding polytope is isomorphic to a reflexive polytope.
A consequence of Remark 2.5, along with the fact that the converse of Remark 2.2 does not hold in general, is that infinite families of inequivalent toric Fano CDPs can be constructed from a single isomorphism class of reflexive polytopes. In fact, Figure 2.2 gives an example of how to construct such a family. Such examples explain the restriction of the statement of Conjecture 1.2 to non-toric CDPs.
For the purposes of classifying Fano CDPs, it would be useful to have some kind of normal form, that is, a distinguished representative in each equivalence class of Fano CDPs. While we have yet to find a natural normal form for Fano CDPs, in the following we describe a number of normalizing assumptions we will be making whenever considering a Fano CDP. This will be key to our strategy for obtaining bounds on the structure of Fano CDPs.
A Fano CDP is normalized if
satisfies the four criteria of Definition 1.1;
If is linear, then it does not have integral slope.
Note that any non-toric Fano CDP is equivalent to a normalized Fano CDP. Indeed, by virtue of being Fano, we can certainly satisfy the first property above by replacing with an equivalent CDP. Furthermore, given any linear with integral slope, we can shear and translate so that the corresponding function becomes zero, and then eliminate it. We repeat this until we have only two functions remaining (and have a toric CDP), or there are no more linear functions with integral slope.
In the remainder of this paper, we will always be dealing with a normalized Fano CDP with base polytope and functions . In particular, there are integers satisfying properties 2 and 3 of Definition 1.1. It is often more convenient to consider the translated functions , which are at height one. Applying Property 2 of Definition 1.1 to the translated functions yields
3. Bounds on Number of Functions
In this section we establish a bound on the number of functions in a normalized Fano CDP that is dependent on the base of the CDP (Theorem 3.9). This bound is established as follows: after assuming , we pick a point so that and its reflection through the origin both lie on the coordinate axis and in the base polytope. We give both lower and upper bounds on the sum
where are the functions in a Fano CDP. The lower bound on (4) follows from Inequality (2). The upper bound on (4) is provided by Lemma 3.7, which uses the concavity of the functions to provide an upper bound for the sum . We sum (4) over all . By arguing that the upper bound on can only be achieved times for fixed , as otherwise would be linear with integral slope (see Lemmas 3.5 and 3.7), the bound given in the theorem is obtained.
We establish some straightforward results on Fano CDPs. We let be a -dimensional normalized Fano CDP with base polytope and translated functions , which are at height one. For simplicity, we assume that . First we introduce the notion of integral and non-integral CDPs:
The function is said to be integral if for all , all points . Otherwise we say that is non-integral.
If is linear, then it is integral in the above sense if and only if it has integral slope.
For any , the function is integral if and only if .
See [IS17, Remark 3.7] ∎
Suppose . Then is linear along the line segment from to for all .
Restrict the base polytope to the segment, which we denote by , and let be a facet of containing the point above the origin and some other point of . Suppose meets another facet, say , along . Extending , its value over the origin would be larger than 1 by the concavity of . Then the point lies between the extension of and the origin, contradicting being at height one. ∎
If is identically one along every coordinate axis, then .
By concavity of , on the convex hull of the intersection of with the coordinate axes. Thus by Lemma 3.4, along the line from the origin to any point , and hence on .
Suppose that there is some point such that . Choosing sufficiently small so that , the concavity of would imply that , a contradiction. Hence . ∎
If is non-integral, then for some . In particular, .
The point is in . Since at height one, there is some such that , that is,
where is the component of . Since by assumption and Lemma 3.3, and , we have the desired result. ∎
Let lie on one of the coordinate axes, and suppose that .
If is non-integral, then
If is integral but non-linear along the line segment from to , then
If is integral and linear along the line segment from to , then
First assume that is non-integral. Then, by the concavity of and by Lemma 3.6, we have
and the result follows.
Suppose is integral. By shearing we may assume that . If is linear along the line segment from to , then , and the result in this case holds. Otherwise suppose is non-linear along this line segment. Then the slope of along the line segment from the origin to is at most , which gives . Since the bound on the sum is invariant under shearing, the result follows. ∎
3.3. Main Result
We now show that the number of functions in a normalized Fano CDP is bounded by a value determined only by the base of the CDP.
Fix a lattice polytope containing the origin in its interior. For any primitive element of the lattice , define
Let be primitive. Any normalized Fano CDP has at most functions which are either non-integral, or non-linear along the line spanned by .
Set . Define
and . Then by Lemmas 3.7 and 3.4, is exactly the number of functions which are either non-integral, or non-linear along the line spanned by . Note that, for each , we have by Lemma 3.7 that This bound works for non-integral because . Using this and the lower bound given in Inequality (2), we have
and hence . ∎
Now, fix a basis of the lattice . We define the constant by
Let be a -dimensional normalized Fano CDP with base polytope . Then has at most functions.
Let be the -dimensional cross-polytope, that is, is the convex hull in of , where are the standard basis vectors. Then using the standard basis, , so by Theorem 3.9 any normalized Fano CDP with base has at most functions.
In fact, this bound is sharp. For , set
and also set
These functions are all at height one, and satisfy Inequality (2), hence they come from a normalized Fano CDP with non-linear functions.
4. Finiteness over Fixed Base
In this section, we establish Main Theorem 1, which says that there are only finitely many equivalence classes of Fano CDPs over a fixed base. We do this by showing in Theorem 4.1 that if we first fix the number of functions in our CDP, there are only finitely many possibilities over the given base. Main Theorem 1 then follows immediately, given Theorem 3.9.
The proof of Theorem 4.1, is an argument which reduces possibilities by considering the regions of linearity of a function. Let be a CDP with base and functions . Each of the functions are piecewise linear and concave. Consequently, the facets of are convex, and so project down to a subdivision of into polytopes. Moreover, as has integral vertices, the vertices of the polytopes in the decomposition are integral. Hence induces a subdivision of the base polytope into a union of finitely many lattice polytopes. We say that a -dimensional polytope in this subdivision is a region of linearity of the function , where is the dimension of the base polytope .
Let be a fixed lattice polytope and a fixed positive integer. Up to equivalence, there are only finitely many Fano CDPs with base polytope and functions.
Suppose is a Fano CDP with base polytope and translated functions . By shearing and by using Inequality (2) we can bound at each point as follows:
Fix regions of linearity.
As discussed above, each induces a subdivision of into its regions of linearity; the set of these regions we denote by . Since contains only finitely many lattice points, there are only finitely many possibilities for the sets , so we may assume that we have fixed them once and for all.
Give upper bounds for .
Consider the polytopes given by intersections of the form , where each . Let be one of the -dimensional polytopes containing the origin obtained through this process. Note that the restriction of each to is a linear function for . Let be the cone generated by the elements of . By [CLS11, Theorem 11.1.9], it contains a lattice basis . Any element of is a scalar multiple of a point in , and hence contains points where for some . As is convex, we can assume that .
The unimodular basis corresponds to a basis in so that . Thus, if we shear using , then stays fixed for . Hence we may assume that for all and .
As the restriction of to is a linear function, it is determined by its values at the points in the set . Let be the extension of this function to all of . By the concavity of , the maximum of bounds by above. Consider the set of linear functions defined on such that for each point either or . There are only finitely many such functions, and the maximum value obtained by them provides an upper bound for and hence for .
Give lower bound for .
The upper bounds for give a lower bound for , through use of Inequality (2), that is, .
Give upper bound for .
Let . Let be sufficiently small so that and the origin are in the same region of linearity of . By the concavity of , the line through the points and provides an upper bound for . Moreover, this upper bound is maximized by increasing the value for and decreasing the value for . Thus the upper bound and the lower bound for given in Step 3 provides an upper bound for .
Give lower bounds for .
The upper bounds for yield lower bounds for , again by Inequality (2).
Having bounded the functions both from above and below, we may now conclude the proof of the theorem:
Conclude finiteness by counting possible vertices of graphs.
As the vertices of the graph of each are integral by definition, and the function values here are bounded above and below, there are only finitely many ways to choose the vertices of . Since is determined by its vertices, there are only finitely many possibilities for the .
Over any fixed base , there are at most finitely many equivalence classes of Fano CDPs.
5. Two-Dimensional Fano -Varieties
In this section, we enumerate the non-toric two-dimensional Fano CDP equivalence classes:
Our enumeration strategy will include choosing preferred representatives for each equivalence class. Throughout this section we assume that CDPs are normalized, as per Definition 2.6. The base must include both and , so by Theorem 3.9, our normalized CDP has at most four functions.
Suppose is a translated function in a normalized two-dimensional Fano CDP. Then either or .
Assume, towards contradiction, that both and .
If is non-integral, then 0 is not a vertex of and thus and lie on the same facet . By the height one restriction there are such that , and ,. Under the constraints of positivity and , we deduce , and the facet is a horizontal line at . By assumption, is non-integral, and so this facet has a non-integral vertex, which is impossible.
Suppose is a translated function in a normalized two-dimensional Fano CDP. If , then . Similarly if , then .
If and , or vice versa, then may be sheared so that both values are positive, contradicting Lemma 5.2. ∎
We can use shearing within an equivalence class to determine a preferred value of the first functions at , and then use these lemmas to deduce additional restrictions. First, we shear the functions so that is non-negative and in a preferred range:
Using Lemma 5.3 we deduce that
We can argue about the final function by using the condition of Inequality (1), namely on . Consequently,
The latter inequality is strict if is in the interior of . Finally, from this, and Lemma 5.3 we conclude that if ,
These bounds severely restrict the possible base polytope for our representatives.
Under the above restrictions on the form of the functions, there are no normalized two-dimensional non-toric Fano CDPs with a base such that .
Suppose are the translated functions of a Fano CDP with such a base . Assume ((E1)), ((E2)), ((E3)), and ((E4)). We can also assume for either or , given Lemma 5.2. By ((E1)) or ((E2)), if there is some such that , then inequality (1), , cannot be satisfied. Hence for all , and so by Lemma 5.2, for all . Combining inequalities (1) and ((E4)) gives from which we deduce that . Thus is toric. ∎
We now fix a normalized two-dimensional non-toric Fano CDP with base . The base polytope thus has at least one of or as boundary points. Without loss of generality, in our choice representative we assume that is a boundary point of . In the following, we further assume that (again, shearing if necessary).
The next parameter that we consider is the length of the base polytope, which we denote by . If , then . As mentioned above, Theorem 3.9 gives the bound . We have listed all two-dimensional Fano CDPs over with three and four functions in Tables 1 and 2, respectively. If , then is an interior point of . The inequality computed in Theorem 3.9 is strict, which yields a bound . In this case (), our normalized CDP has exactly three functions .
Suppose that has length larger than two. If is integral, then has at most one interior vertex.
It follows directly from Lemma 3.4 that the only vertex must live above . ∎
Suppose that . For , if is non-integral, then up to shearing, either
has a single vertex at some integer satisfying :
or is simply a line with slope :
We have determined that and by Lemma 3.6 there is a positive integer so that . Either is a line, or the point is a vertex. If it is not a line, then on to ensure that the facet is at height one. ∎
The shape of can similarly be described. The normalization of the other functions gives that . Furthermore, recall that (by ((E3))).
Next we consider the possible values at the boundary of the polytope, . The major restricting factor is that the regions of linearity start and end at lattice points, and the functions remain at height one. We next show that the nature of the equations limits to be 3, 4 or 6, and with this information we can construct all cases by iterating through possible choices of shape, and for the three functions.
To bound we look closer at the inequalities. These lemmas determine the possible values for , , and . For , either for some which divides , or for some non-negative integer .
The permissible values for can be found by subtracting from the permissible values of . Moreover, we only keep the values that imply and deduce
for some and . We substitute these possibilities into Equation (3), and search for integer solutions to
We find a finite number of integral solutions to this equation, and find that in these solutions, . We illustrate one of these computations in the next example. The equivalence classes of Fano CDPs with three functions and base of length , , and are given in Tables 3, 4, and 5 respectively. This completes the proof of Theorem 5.1 (and Main Theorem 2).
Suppose that , , and . We seek integer solutions to the equation
Simplifying to , it is easy to see that either or . Without loss of generality, assume . With simple algebra it follows that . Thus or . With the values for and fixed, we are able to solve for . The resulting solutions are .