Projective geometry of Wachspress coordinates
Kathl\'en Kohn, Kristian Ranestad
TL;DR
This paper explores the algebraic and geometric properties of Wachspress coordinates for convex polytopes, introducing a unique hypersurface related to the polytope's non-faces and connecting it to classical algebraic geometry.
Contribution
It generalizes Wachspress's construction of adjoint curves to higher dimensions, linking it with the adjoint polynomial and algebraic geometry concepts.
Findings
Identifies a unique hypersurface passing through non-faces of a polytope.
Describes the Wachspress map and variety, including their algebraic properties.
Classifies polytopes in 2D and 3D with special hypersurfaces, finding elliptic and K3 surfaces.
Abstract
We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the…
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Figure 22| comb. | facet | ||||||
| type | sizes | (deg., sec. genus) | (deg., sec. genus) | (deg., sec. genus) | |||
| minimal K3 | |||||||
| (smooth quartic in ) | |||||||
| line | minimal K3 | ||||||
| twisted cubic curve | minimal K3 | ||||||
| quadric surface | non-minimal K3 | ||||||
| del Pezzo surface in | non-minimal K3 | ||||||
| Fano -fold in | rational scroll in | non-minimal K3 | |||||
| rational elliptic surface in | minimal elliptic | ||||||
| elliptic K3-surface in | minimal elliptic | ||||||
| K3-surface in | non-minimal K3 | ||||||
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Projective geometry of Wachspress coordinates
Kathlén Kohn
Kristian Ranestad
Abstract
We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves¡ are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.
1 Introduction
Barycentric coordinates on convex polytopes have many applications, such as mesh parameterizations in geometric modelling, deformations in computer graphics, or polyhedral finite element methods. Whereas barycentric coordinates are uniquely defined on simplices, there are different versions of barycentric coordinates on more general convex polytopes. For instance, mean value coordinates and Wachspress coordinates are both commonly used in practice. A nice overview on different versions of barycentric coordinates, their history and their applications is [4].
This article provides an algebro-geometric study of Wachspress coordinates, which were first introduced for polygons by Wachspress [11] and later generalized to higher dimensional convex polytopes by Warren [13]. Wachspress defined the adjoint curve of a polygon as the minimal degree curve passing through the intersection points of pairs of lines containing non-adjacent edges of the polygon; see Figure 1. Warren defined the adjoint polynomial for any convex polytope in : he first fixes a triangulation of into simplices such that the vertices of each occurring simplex are vertices of , and then defines the adjoint to be the polynomial
[TABLE]
where , the set of vertices of a polytope is denoted by , and denotes the linear form associated to a vertex . Warren shows that the adjoint is independent of the chosen triangulation, so we set . Moreover, he observes that the adjoint of vanishes on the intersections of pairs of hyperplanes containing non-adjacent facets of the dual polytope . If is a polygon, this implies that coincides with Wachspress’ adjoint of the dual polygon . Finally, Warren defines the Wachspress coordinates of a polytope as follows:
[TABLE]
where denotes the set of facets of , denotes the facet of the dual polytope corresponding to the vertex , and denotes the vertex of the dual polytope corresponding to the facet .
Warren’s adjoint also appears in algebraic statistics and intersection theory.
Intersection theory.
The adjoint is the central factor in Segre classes of monomial schemes. This was observed by Aluffi [1, 2], but without the language of adjoints. To explain Aluffi’s setup, we consider a smooth variety with smooth hypersurfaces meeting with normal crossings in . For an index tuple , we write for the hypersurface obtained by taking with multiplicity . Any finite subset of index tuples defines a monomial subscheme as well as a Newton region which is the complement in of the convex hull of the positive orthants translated at the points in , i.e., . From Aluffi’s results [1, 2] we deduce the following in Section 2:
Proposition 1.1**.**
If the Newton region is finite, the Segre class of the monomial subscheme in the Chow ring of is
[TABLE]
Remark 1.2**.**
If has vertices at infinity, these are points at infinity in the direction of the standard basis vectors . The Segre class of is the limit of (1.3). This limit is a rational function of the same form as (1.3) where the linear form associated to a vertex at infinity in the direction of is ; see Remark 2.7.
Example 1.3**.**
We are using the running example in [1]. Here and . The Newton region is a hexagon with a vertex at infinity in the direction of the second standard basis vector. We will see in Remark 2.7 that the Segre class of the monomial scheme is
[TABLE]
[TABLE]
Algebraic statistics.
Here the adjoint appears when studying the moments of the uniform probability distribution on a polytope. For a probability distribution on and an index tuple , the -th moment of is
[TABLE]
If is the uniform probability distribution on a polytope , we shortly write . A suitably normalized generating function over all moments of a fixed simplicial polytope in with vertices is a rational function whose numerator is Warren’s adjoint of [10]:
[TABLE]
In [10, p. 6] it was also observed that the adjoint of the simplicial polytope vanishes on all linear spaces which do not contain faces of the dual polytope but are intersections of hyperplanes spanned by facets of . Moreover, the authors conjectured that the adjoint is uniquely characterized by this vanishing property. In this article, we prove this conjecture.
Unique adjoint hypersurfaces.
We generalize Wachspress’ construction of the adjoint from polygons to polytopes and show that our definition of the adjoint coincides with Warren’s definition. In particular, we answer a question by Wachspress [12]: in 2011 he asked for a geometric construction of the unique adjoint associated to a tesseract (i.e., a four-dimensional hypercube); see Example 1.19.
To avoid dealing with degenerate situations such as parallel faces, we typically work in complex projective -space . With a polytope in we denote the collection of the Zariski closures of the faces of a convex polytope in . If the polytope is full-dimensional, the union of these projective subspaces is a hyperplane arrangement . We study the residual arrangement of , which consists of all linear spaces that are intersections of hyperplanes in and do not contain any face of .
Example 1.4**.**
If is a polygon in the plane with edges, the residual arrangement consists of points. These are exactly the intersection points of the extended edges outside of the polygon , as studied by Wachspress.
If is a triangular prism (i.e. the second polytope in Table 1), the planes spanned by the two triangle facets of intersect in a line and the planes spanned by the three quadrangle facets intersect in a point. The residual arrangement consists of this line and this point.
If is a cube, each pair of planes spanned by opposite facets of the cube intersects in a line. Hence, the residual arrangement consists of three lines. If is a regular cube, the three lines lie in a common plane (the “plane at infinity”). If the cube is perturbed, the three lines are skew.
The concept of residual arrangements allows us to generalize Wachspress’ adjoint. To be more precise we prove the following assertion in Section 2 (see Theorem 2.1).
Theorem 1.5**.**
Let be a full-dimensional polytope in with facets. If the hyperplane arrangement is simple (i.e. through any point in pass at most hyperplanes in ), there is a unique hypersurface in of degree which vanishes along the residual arrangement .
If is simple, we call the adjoint of the polytope . Another way of stating this theorem is: for a polytope with a simple hyperplane arrangement , the ideal sheaf of the residual arrangement twisted by has a unique global section up to scaling with constants, i.e. .
Example 1.6**.**
If is a polygon in the plane, then the adjoint is exactly the adjoint curve described by Wachspress. If is a triangular prism, then the adjoint is the plane spanned by the line and the point in the residual arrangement .
If is a perturbed cube such that the three lines in the residual arrangement are skew, then the adjoint is the unique quadric passing through the three lines. If the cube is regular and the three lines lie in a common plane, then the adjoint degenerates to that plane doubled. However, in this case the plane arrangement is not simple and there is a three-dimensional family of quadrics passing through the three lines, as such a quadric consists of the common plane of the three lines together with any other plane.
To compare Warren’s adjoint with our definition, we recall Warren’s vanishing observation stated above: his adjoint of a polytope vanishes on the codimension two part of the residual arrangement of the dual polytope [13, Thm. 5]. We generalize this assertion to the whole residual arrangement (for a proof, see Section 2).
Proposition 1.7**.**
For a full-dimensional convex polytope in , Warren’s adjoint vanishes along the residual arrangement of the dual polytope . In particular, for a simple hyperplane arrangement , the zero locus of the adjoint polynomial is the adjoint hypersurface .
This shows that, for every convex polytope in with facets, there is a hypersurface of degree passing through the residual arrangement (namely the zero locus of ). If the hyperplane arrangement in is simple, this hypersurface is unique due to Theorem 1.5. Otherwise, this hypersurface might not be unique as Example 1.6 demonstrates. Nevertheless, the adjoint hypersurface is well-defined, independently of the simplicity of the hyperplane arrangement: we can deduce from Proposition 1.7 that the adjoint hypersurface of a polytope with a non-simple hyperplane arrangement is the unique limit of the adjoint hypersurfaces of all perturbations of with a simple hyperplane arrangement (for a proof, see Section 2).
Corollary 1.8**.**
Let be a full-dimensional polytope in with facets. If and are continuous families of polytopes with facets and simple hyperplane arrangements for such that and , then the limits and of their adjoint hypersurfaces coincide.
Remark 1.9**.**
The unique limit in Corollary 1.8 is the adjoint hypersurface of , independently of the simplicity of the hyperplane arrangement . In a real affine chart where is convex, this is the (possibly non-reduced) zero locus of the adjoint polynomial .
The Wachspress coordinate map.
The study of the map defined by the Wachspress coordinates (1.2), particularly in dimensions two and three, was proposed by Garcia and Sottile [6] and Floater and Lai [5]. The two-dimensional case was thoroughly analyzed by Irving and Schenck [9]. We extend their results to three and higher dimensions. In particular, we will see that already the three-dimensional case is considerably more complicated than the case of polygons.
The dual polytope of a simple polytope in is simplicial, so the facet of corresponding to a vertex of is a simplex. Thus, the Wachspress coordinates (1.2) reduce in this case to
[TABLE]
The linear form simply is the defining equation of the hyperplane spanned by the facet . As the polytope is simple, the numerator of each rational function in (1.4) has degree , where denotes the number of facets of . Hence, these numerators define a rational map
[TABLE]
where denotes the number of vertices of and is a homogeneous linear equation defining the projective closure of the hyperplane spanned by the facet . We call the Wachspress map of the simple polytope , and call the coordinates of enumerated by the vertices the Wachspress coordinates. The Wachspress map is not defined everywhere: if we assume the hyperplane arrangement to be simple, it turns out that is undefined exactly along the residual arrangement of (for a proof see Section 3).
Theorem 1.10**.**
For a full-dimensional polytope in with a simple hyperplane arrangement , the base locus of the Wachspress map is the residual arrangement .
This assertion shows that the homogeneous forms in (1.5) live in the space
[TABLE]
Evaluating the homogeneous form corresponding to a vertex at a vertex yields zero if and only if . Hence, the homogeneous forms in (1.5) are linearly independent in . In fact, they form a basis of , as we show in Section 2 (see Theorem 2.8).
Theorem 1.11**.**
For a full-dimensional polytope in with a simple hyperplane arrangement , the dimension of equals the number of vertices of .
Hence, the Wachspress map is in fact of the form . Its image is the Wachspress variety of the polytope . The projective span of the image of the unique adjoint under the Wachspress map is a projective subspace of . We denote the linear projection from this subspace by . In Section 3, we show that this projection restricted to the Wachspress variety is the inverse of the Wachspress map.
Theorem 1.12**.**
Let be a full-dimensional polytope in with vertices and a simple hyperplane arrangement . The dimension of is . The projection from restricted to the Wachspress variety is the inverse of the Wachspress map .
As the Wachspress map is rational, we consider the blow-up of along the base locus of (i.e., along the residual arrangement according to Theorem 1.10). Thus is the closure of the graph of the Wachspress map . In this way, the projection of onto the second factor yields a lifting of the Wachspress map. We summarize our maps in the commutative diagram in Figure 2 (we introduce the blowup after Proposition 1.15):
Example 1.13** (see [9]).**
When is a -gon in , the Wachspress coordinates span a -dimensional vector space and the residual arrangement consists of points, so is the blowup of these points and the morphism maps into The image is a surface of degree . The adjoint curve is, for a general , a smooth curve of degree whose image under has degree .
We extend these results by Irving and Schenck to polytopes in -space.
Example 1.14**.**
If is a tetrahedron, the residual arrangment is the empty set, the adjoint surface is empty and the Wachspress map is the identity map on .
If is a triangular prism, the adjoint surface is the unique plane spanned by the line and the point of (see Example 1.6). The Wachspress map is given by , where the and denote the defining equations of the planes spanned by triangular and quadrangular facets, respectively. The image of the Wachspress map is the Segre embedding of in . The restriction of the Wachspress map to the adjoint plane is the projection from the point, so the image of the adjoint plane is a line.
If is a cube, perturbed so that the plane arrangement is simple, the adjoint surface is the unique quadric surface that contains the three lines of (see Example 1.6). The Wachspress map is given by , where and denote the defining equations of two planes spanned by opposite facets of the perturbed cube . The image of the Wachspress map is the Segre embedding of in . The restriction of the Wachspress map to the adjoint surface contracts the ruling of lines that meet all three lines in , so the image of the surface is a twisted cubic curve.
In fact, for all other three-dimensional polytopes defined by simple plane arrangements, the image of the adjoint surface under the Wachspress map is a surface. We prove the following extension of the results by Irving and Schenck in Section 3.
Proposition 1.15**.**
Let be a full-dimensional polytope in with facets and a simple plane arrangement . Let be the number of isolated points in the residual arrangement , let be the number of double points (i.e. points where exactly two lines in meet), and let be the number of triple points (where three lines in meet). The rational variety has degree
[TABLE]
and sectional genus
The image of the adjoint surface is a curve if and only if is a triangular prism or a cube (see Example 1.14). If is not a triangular prism, cube, or tetrahedron, the image is a surface of degree
[TABLE]
and sectional genus
The image of any surface in linearly equivalent to the strict transform of has degree
[TABLE]
while the sectional genus is
[TABLE]
As the variety is typically not smooth (see Lemma 3.4), we blow up a little further. The map (see Figure 2) also blows up along , but decomposed into a sequence of blowups each along a smooth locus.
To be precise, we consider a decomposition , where is the union of the irreducible components of of codimension that are not contained in any component of codimension in . Similarly, we decompose the singular locus , where is the union of codimension linear spaces in the singular locus of that lie on hyperplanes in . We first blow up all isolated points and [math]-dimensional singular points in , i.e . Next, we blow up the strict transform of the lines , and so on. The procedure ends with the blowup of the strict transform of the codimension three locus and finally of the strict transform of the codimension two locus . The map is the composition of these blowups, which ensures that is smooth.
Polytopal hypersurfaces.
The blow-up allows us to relate adjoints of polytopes to the classical notion of adjoints of hypersurfaces and divisors. For a hypersurface of degree , an adjoint is a hypersurface of degree that contains every component of codimension of the singular locus of where has multiplicity . More generally, for a smooth variety and a smooth subvariety of codimension one, an adjoint to in is a codimension-one subvariety of whose class in the Chow ring of is (i.e., the canonical class of plus the class of ). This adjoint appears in the adjunction formula: the canonical class of is restricted to . Therefore adjoints play an important role in the classification of varieties (for a nice survey, see [3]).
If the strict transform of in is linearly equivalent to a smooth divisor , then we can compare adjoints of in to the adjoint of the polytope . More formally, we study the linear system of divisors in which have degree and vanish with multiplicity along . We call the hypersurfaces in the polytopal hypersurfaces associated to . Note that the hyperplane arrangement is a polytopal hypersurface. In Section 4, we show that each smooth strict transform in of a polytopal hypersurface associated to a polytope with a simple hyperplane arrangement has a unique adjoint in . The hypersurface of the polytope is an adjoint for any that is smooth outside , and its strict transform is the main component of the unique adjoint of in (for a proof see Section 4).
Proposition 1.16**.**
Let be a full-dimensional polytope in with a simple hyperplane arrangement . Moreover, let be the strict transform of the adjoint under the blow-up , and let be its class in the Chow ring of . Every strict transform in of a polytopal hypersurface satisfies
[TABLE]
where is an effective divisor whose image is contained in the singular locus of . If is smooth, it has a unique adjoint in , and thus a unique canonical divisor: namely .
As a smooth strict transform of a polytopal hypersurface has a unique canonical divisor, it is interesting to determine its birational type. In Section 4, we identify all combinatorial types of simple polytopes in dimensions two and three which admit smooth strict transforms of their polytopal hypersurfaces and we describe their birational types.
Proposition 1.17**.**
Let , and let be a general polygon in with edges. There is a polygonal curve with a smooth strict transform in if and only if . These curves are elliptic.
Theorem 1.18**.**
Let be a combinatorial type of simple three-dimensional polytopes, and let be a general polytope in of type . There is a polytopal surface with a smooth strict transform in if and only if is one of the nine combinatorial types in Table 1.
In that case, the general is either an elliptic surface (if has a hexagonal facet) or a K3-surface (if has at most pentagonal facets).
We describe all combinatorial types of simple three-dimensional polytopes with smooth strict transforms of their polytopal surfaces in detail in Subsections 4.1 and 4.2. Some of their key properties are summarized in Table 1.
Since Wachspress asked for a geometric construction of the unique adjoint associated to a tesseract (i.e., a four-dimensional hypercube) in 2011 [12], we discuss this higher dimensional example in detail.
Example 1.19**.**
Let be a hypercube in , which is perturbed so that its hyperplane arrangement is simple. The polytope has facets and vertices. The residual arrangement consists of subspaces of codimension two, as every pair of hyperplanes spanned by opposite facets of intersects in such a subspace. There is an -dimensional family of lines in passing through all subspaces in . The union of these lines forms the adjoint hypersurface of degree .
Similar to the cube case (cf. Example 1.14), we see that the Wachspress variety is the Segre embedding of in . The restriction of the Wachspress map to the adjoint hypersurface contracts each line in the ruling described above. Explicit calculations for suggest that the image of under the Wachspress map is a rational variety of dimension projectively equivalent to the image of a general -dimensional subspace of . In fact, we computed for small that the intersection of with two general hyperplanes which contain the image of has two components: both are rational and isomorphic to the image of mapped by hypersurfaces of degree that contain linear spaces of dimension in general position (these linear spaces are the intersection of the with ). The intersection of the two components is an anticanonical divisor on each of them.
The general hyperplane sections of are images of hypersurfaces of degree that contain , while the polytopal hypersurfaces in have degree and are singular along . So has an irreducible polytopal hypersurface and its images in is the intersection with a quadric hypersurface. In particular, that image is a Calabi-Yau variety.
It is natural to ask if all our results can be extended to polytopes in dimension four and higher. Moreover, since many of our results are restricted to polytopes which are defined by simple hyperplane arrangements, it would be interesting to study more general classes of polytopes.
Problem 1**.**
Can the results in Proposition 1.15, Theorem 1.18 and Table 1 be extended to polytopes in whose hyperplane arrangements are simple, with ?
In particular, is there a simple description of such polytopes which have smooth strict transforms of their polytopal hypersurfaces? What are their birational types?
Problem 2**.**
Can our results be extended to polytopes whose hyperplane arrangements are not simple?
In particular, find a general definition of the adjoint hypersurface of a polytope which does not depend on the simplicity of its hyperplane arrangement and does not involve limits as in Corollary 1.8.
Remark 1.20**.**
Let be one of the combinatorial types in Table 1 and let be a general polytope in of type . We verified computationally that the rational variety is arithmetically Cohen Macaulay. This further extends results by Irving and Schenck [9], who showed in the case of polygons in the plane that the Wachspress surface is arithmetically Cohen Macaulay.
Problem 3**.**
Does Remark 1.20 hold for every combinatorial type of simple three- dimensional polytopes? What about simple polytopes in dimensions, and non-simple polytopes?
2 The adjoint
Our main result in this section is to show that every full-dimensional polytope defined by a simple hyperplane arrangement has a unique adjoint .
Theorem 2.1**.**
Let with . For every full-dimensional polytope in with facets and a simple hyperplane arrangement , there is a unique hypersurface in of degree vanishing on , i.e. . Moreover, this hypersurface does not pass through any vertex of , and .
We prove this assertion by induction on and . To start the induction, we need to consider the cases (i.e. is a simplex) and . As there are no polytopes with more than two facets in , the case follows from the case where is a simplex.
Lemma 2.2**.**
Theorem 2.1 holds if is a simplex, i.e. if .
Proof.
If is a simplex, its residual arrangement is empty. So . This shows that and . Moreover, the up to scalar, unique nonzero global section of is a constant function on , which does not vanish on any vertex of . ∎
For the induction step, we assume . We pick any hyperplane in the arrangement and consider the polytope obtained by removing the facet corresponding to from . To relate the residual arrangements of and , we first state several combinatorial observations:
Remark 2.3**.**
The facet corresponding to is an -dimensional polytope. The faces of which are not faces of are exactly the faces of . 2. 2.
We denote by the number of facets of . These facets are the intersections of with hyperplanes in . The intersections of with the other hyperplanes in are contained in the residual arrangement (and not in ). 3. 3.
The hyperplane cuts off faces of , which are no longer faces of . This could be a single vertex, or up to several faces of codimension at least two. We denote the union of all subspaces corresponding to cut-off-faces by . All subspaces in are intersections of the hyperplanes . Note that the set of cut-off faces has the structure of a polyhedral complex whose -skeleton is connected, unless it is a single vertex. 4. 4.
The subspaces of codimension two in are exactly the subspaces of codimension two in together with the subspaces described in item 2 of this remark and the subspaces of codimension two in the cut-off part . In particular, every codimension-two subspace in is contained in . 5. 5.
More generally, for , there can be subspaces of codimension which are contained in either one of the two residual arrangements of and , and not in the other. Such subspaces which are contained in but not are either contained in the cut-off part or are the intersection of a common face of and with the hyperplane . Codimension- subspaces in which do not appear as subspaces of codimension in are contained in a higher-dimensional subspace in the cut-off part .
Corollary 2.4**.**
The arrangement obtained from by intersecting every irreducible component of with the hyperplane consists of subspaces of codimension one in as well as the residual arrangement of the facet inside of .
Proof.
The hyperplanes in are the ones described in Remark 2.3.2. In addition, consists of the intersection of with subspaces in which are intersections of . These subspaces do not contain faces of , so they do not contain faces of and lie in the residual arrangement .
For the other direction, we consider a maximal subspace in . If would contain a face of , this face would also be a face of due to Remark 2.3.1. As is contained in , we see that the hyperplane contains a face of , which contradicts our assumption that the hyperplane arrangement is simple. Hence, is contained in a subspace of . This subspace is either itself, or is the intersection of this subspace with . ∎
With this, we consider the map which restricts a polynomial in the ideal of the residual arrangement to the hyperplane :
[TABLE]
Note that our goal is to show that has a unique global section up to scalar, whose zero locus is the adjoint . The map (2.1) is surjective. By determining its kernel, we can extend it to a short exact sequence:
Proposition 2.5**.**
The map (2.1) can be extended to the following short exact sequence:
[TABLE]
where the left map is the multiplication with the defining equation of .
Proof.
The kernel of the map (2.1) consists of functions which vanish on and on . Hence, we need to determine the parts in which are not contained in . Due to Remark 2.3.3, the subspaces in the cut-off part are contained in but not in . Remark 2.3.4 and 2.3.5 show that subspaces in are either contained in or in subspaces in . As we assume the hyperplane arrangement to be simple, the subspaces in are not contained in . Thus, consists of subspaces in which are not contained in . Finally, Remark 2.3.4 and 2.3.5 imply that all subspaces in which do not lie in are contained in . ∎
Now we conclude the proof of Theorem 2.1, by computing the cohomology of this short exact sequence.
Proposition 2.6**.**
We have and .
Proof.
By applying our induction hypothesis on , we see that . So has a unique adjoint . Again by the induction hypothesis, this adjoint does not pass through any vertices of . Since the cut-off part contains at least one vertex, we deduce that .
For the second part, we consider the embedding and the following short exact sequence together with the dimensions of the first cohomology groups:
[TABLE]
In this diagram, we denote by the dimension of the space of global sections of . To conclude the proof of Proposition 2.6, we need to show that .
From the analogous short exact sequence for and our induction hypothesis that and , we deduce that the space of global sections of the pushforward of to has dimension . Now we pick any vertex in the cut-off part . As this vertex is not contained in the residual arrangement , we see that the space of global sections of the pushforward of to has dimension .
If the cut-off part consists only of this vertex , we are done. Otherwise, we pick a cut-off edge such that is one of its two vertices. The line is the intersection of hyperplanes in . There are two more hyperplanes in which define the two vertices of the edge . The other hyperplanes in intersect the line in points which are contained in subspaces in the residual arrangement . Hence, functions on of degree are uniquely determined by functions of this degree on , i.e. the space of global sections of the pushforward of to also has dimension .
By Remark 2.3.3, the cut-off edges are connected. By repeating the same argument as above, functions of degree on , where denotes the union of all lines corresponding to cut-off edges, are uniquely determined by functions of this degree on . Thus, the space of global sections of the pushforward of to has dimension as well.
If , we are done. Otherwise, we prove by induction on that functions of degree on , where denotes the union of all linear subspaces corresponding to -dimensional cut-off faces, are uniquely determined by functions of this degree on . As , we have proven the induction beginning above. For the induction step, we assume and consider an arbitrary cut-off face of dimension . This face is defined by hyperplanes in . The other hyperplanes in intersect in -dimensional subspaces. Some of these subspaces are facets of , and our induction hypothesis yields that functions of degree on extend uniquely to these facets. The other hyperplanes in are already contained in subspaces in the residual arrangement . A general line in the face intersects the hyperplanes in in points. Since , functions of degree on extend uniquely to the whole line . As was chosen generally, functions of degree on are uniquely determined by functions of this degree on . This shows that the space of global sections of the pushforward of to has dimension . Thus, and Proposition 2.6 is proven. ∎
Proof of Theorems 1.5 and 2.1.
Theorem 2.1 implies Theorem 1.5. We prove Theorem 2.1 by induction on and . The induction beginning, i.e. or , follows from Lemma 2.2. For the induction step, we assume and , and we use the same notation as introduced in Remark 2.3.
We first prove that has a unique adjoint. By applying our induction hypothesis to the facet , we see that , meaning that has a unique adjoint as a polytope in . From Corollary 2.4 we deduce : any nonzero section of vanishes on hyperplanes in as well as the adjoint of , so these sections are all proportional. Similarly, the induction hypothesis and Corollary 2.4 show . Hence, Propositions 2.5 and 2.6 show that , i.e. the polytope has a unique adjoint , and that .
Finally, we assume for contradiction that this adjoint passes through a vertex of . We then pick any hyperplane in containing and repeat the above argument (including the short exact sequence) with this specific hyperplane . Due to Corollary 2.4, the intersection of the adjoint and the hyperplane consists of (several) linear subspaces of codimension one in and of the adjoint of the facet . By applying our induction hypothesis to this facet, we see that the adjoint of the facet does not pass through any vertex of . Hence, one of the linear subspaces of codimension one has to contain . This contradicts our assumption that the hyperplane arrangement is simple, since is a vertex of the facet and, by Remark 2.3.2, is also contained in one of the hyperplanes in not defining this facet. ∎
Now we have generalized Wachspress’ notion of adjoints of polygons to polytopes defined by simple hyperplane arrangements. Next we show that Warren’s adjoint polynomial of a polytope vanishes along the residual arrangement of the dual polytope . This shows that our notion of the adjoint hypersurface of coincides with Warren’s adjoint of if the polytope is defined by a simple hyperplane arrangement.
Proof of Proposition 1.7.
We proceed analogous to Warren’s proof of the similar statement that vanishes on the codimension two part of [13, Thm. 5]. We use the following equation derived by Warren [13, Thm. 3]: for any linear function ,
[TABLE]
where the set of vertices of is , the set of facets is , the vertex of the dual polytope corresponding to a facet is denoted by , and .
Now we use induction on to prove Proposition 1.7. If , the polytopes and are line segments, so the residual arrangement is empty. If , we consider an arbitrary irreducible component of of codimension . This component is the intersection of hyperplanes in . These hyperplanes correspond to vertices of the polytope . That the intersection of the hyperplanes does not contain a face of is equivalent to the fact that the vertices do not span a face of .
We show that the adjoint polynomial vanishes on , by proving that, for each linear function , each summand in the right hand side of (2.2) vanishes on . Thus, we fix an arbitrary facet of . If at least one of the vertices is not contained in , say , then the summand in the right hand side of (2.2) corresponding to contains the term . As this term is the defining equation of the hyperplane , it vanishes on . Hence, we only have to consider the case that all vertices are vertices of the facet . This means that the hyperplanes contain the vertex , i.e. . Note that is the hyperplane defining the facet of which corresponds to the vertex of . As the vertices do not span a face of , the intersection of these hyperplanes does not contain a face of the dual polytope . So is contained in the residual arrangement . By the induction hypothesis, Warren’s adjoint polynomial of the polytope (considered as a full-dimensional polytope in the hyperplane of spanned by ) vanishes on . As a function, we can concatenate this polynomial with the projection from the vertex . This yields the adjoint in (2.2). Hence, vanishes on .
All in all, we have shown that the adjoint polynomial vanishes on the residual arrangement . When the hyperplane arrangement is simple and denotes the number of hyperplanes in , we see from Theorem 1.5 that is, up to scalar, the unique polynomial of degree vanishing on , so its zero locus is . ∎
From this we deduce that a polytope with a non-simple hyperplane arrangement also has a unique adjoint: it is the unique limit of the adjoint hypersurfaces of small perturbations of the polytope.
Proof of Corollary 1.8.
Passing to an affine chart, we may consider as a convex polytope in . We may further assume that and are families of convex polytopes as well. By Proposition 1.7, their unique adjoint hypersurfaces and are the zero loci of the adjoint polynomials and , respectively.
Since the family is obtained by continuously moving the vertices of the polytopes in the family, we can triangulate the polytopes consistently, i.e., such that the triangulations move continuously with the family. Indeed we can choose triangulations that do not introduce any new vertices such that the vertices of each simplex are vertices of the respective triangulated polytope. The resulting family of triangulated simplicial polytopes converges for to a triangulation of . From (1.1), we see that the adjoint polynomials converge to . Similarly, by picking some consistent triangulation of the family , we deduce that . Hence, both families of adjoint hypersurfaces and converge to the zero locus of . ∎
Next we show that Warren’s adjoint is the central factor in Segre classes of monomial schemes.
Proof of Proposition 1.1.
According to [2, Thm. 1.1], the Segre class of is
[TABLE]
By [2, Lem. 2.5], the integral (2.3) over a simplex with finite vertices is where . So if we pick any triangulation of the Newton region that does not introduce any new vertices, we see that the Segre class of is
[TABLE]
which proves the assertion. ∎
Remark 2.7**.**
If the Newton region has vertices at infinity in the direction of some standard basis vectors, we can do a similar computation as in (2.4). For a simplex with such infinite vertices, we consider the finite simplex obtained by projecting along its infinite directions. We denote by the normalized volume of . By [2, Lem. 2.5], the integral (2.3) over a simplex with finite vertices and infinite vertices in the direction of some standard basis vectors is , where for the vertex at infinity in the direction of the -th standard basis vector. Repeating the computation in (2.4), we derive that the Segre class of is
[TABLE]
Since the adjoint is well-behaved under limits, it makes sense to define the adjoint of the unbounded Newton region in that way.
Using arguments similar to those in the proof of Theorem 2.1, we show that the dimension of the space equals the number of vertices of the polytope .
Theorem 2.8**.**
Let with . For every full-dimensional polytope in with facets, vertices and a simple hyperplane arrangement , we have that .
Proof of Theorems 1.11 and 2.8.
Theorem 2.8 implies Theorem 1.11. We prove Theorem 2.8 again by induction on and . To start the induction, we consider the case that is a simplex, i.e. . As in the proof of Lemma 2.2, we see that . This shows , which equals the number of vertices of . Since there are no polytopes with more than two facets in , we have also proven the assertion for the case .
For the induction step, we assume and . As before, we pick a hyperplane in and consider the polytope obtained by removing the facet corresponding to from . In the following we will use the same notation as introduced in Remark 2.3. Viewing as a polytope in with facets, the induction hypopthesis yields that is the number of vertices of . By Corollary 2.4, we have . In the following, we show that
[TABLE]
where denotes the number of vertices of and denotes the number of vertices in the cut-off part . Note that the number of vertices of equals . Hence, tensoring the short exact sequence in Proposition 2.5 by , we see that (2.5) and (2.6) imply Theorem 2.8.
We first prove (2.5). By the induction hypothesis, . Evaluating the Wachspress coordinate of corresponding to a vertex at a vertex yields zero if and only if . Thus the vertices of are mapped to linearly independent points by the coordinates of . Hence the vertices in any subset of impose independent conditions on forms of degree that vanish on . In particular, since , we have .
Now we show by induction on that polynomials of degree which vanish on also vanish on , where denotes the union of all linear subspaces corresponding to -dimensional cut-off faces. As , the induction beginning is trivial. For the induction step, we proceed similarly to the proof of Proposition 2.6. We consider an arbitrary cut-off face of dimension . This face is defined by hyperplanes in . The other hyperplanes in intersect in -dimensional subspaces. Some of these are facets of , on which global sections of vanish due to the induction hypothesis, and the others are already contained in subspaces in . Hence, polynomials of degree which vanish on also vanish on these hyperplanes in . Since , such polynomials also vanish on . This proofs the induction step and thus (2.5).
Finally, we show (2.6). As in the proof of Proposition 2.6, we consider the following short exact sequence together with the dimensions of the first cohomology groups:
[TABLE]
Here denotes the dimension of the space of global sections of . To conclude the proof of (2.6) and thus of Theorem 2.8, we show that .
We first observe that is exactly the number of vertices in . Clearly, there are functions of degree defined on these vertices. Let us now consider an arbitrary face of dimension which is either contained in or in . This face is defined by hyperplanes in . Thus, there are vertices in which are defined by the other hyperplanes in . All of these vertices are vertices in . A function defined on these vertices extends uniquely to a function of degree on , and each function of this degree on is uniquely determined by its values on these vertices. As the face was chosen arbitrarily, we have shown that . Hence, we have proven (2.6) and Theorem 2.8. ∎
3 The Wachspress coordinate map
The Wachspress coordinates, see (1.5), are enumerated by the vertices of . For a simple polytope in with facets and a vertex , the coordinate
[TABLE]
vanishes on the hyperplanes in that do not contain :
[TABLE]
Remark 3.1**.**
If is a hyperplane spanned by a facet of , then only the Wachspress coordinates of vertices in do not vanish on . By Corollary 2.4, if the polytope has facets, the coordinates of vertices in have a common factor vanishing on subspaces of codimension one in . The residual factors are precisely the Wachspress coordinates of .
We now show that the common zero locus of the Wachspress coordinates, their base locus, is the residual arrangement .
Proof of Theorem 1.10.
We first show this assertion set-theoretically. To begin with, we consider an irreducible component of the residual arrangement . It is the intersection of hyperplanes in the arrangement . For every vertex of , at least one of these hyperplanes does not contain . Thus, the Wachspress coordinate vanishes along . This shows that the residual arrangement is contained in the base locus of the Wachspress coordinates.
For the other direction, we show by induction on that the base locus of the Wachspress coordinates is contained in the residual arrangement . For the induction beginning, i.e. , the polytope is a line segment and the common zero locus of its two Wachspress coordinates is empty. For the induction step, we assume and consider a point in the base locus of the Wachspress coordinates. As each coordinate vanishes on a union of hyperplanes in , the base locus is contained in . So the point lies in a hyperplane in . Since also the Wachspress coordinates of vertices of in vanish on , the point must in fact lie on at least one other hyperplane in . If there is another hyperplane in containing such that the intersection is in the residual intersection , we are done. Otherwise each hyperplane containing is either or defines a facet of the facet of in . In that case, the point is also contained in the base locus of the Wachspress coordinates of viewed as a polytope in . By our induction hypothesis, is contained in the residual arrangement and thus also in . Hence, we have shown that the base locus of the Wachspress coordinates of equals set-theoretically.
To show that the equality also holds for the scheme, i.e. that the base locus has no embedded components, we let now be any point in and consider the union
[TABLE]
of linear spaces in that pass through the point . We first prove two claims which show how the linear equations of the hyperplanes in yield defining equations of .
Claim 1**.**
If and are linear spaces through that are intersections of hyperplanes in , then the linear span of is either the whole ambient space or it is also an intersection of hyperplanes in . In particular, the span of the union of any collection of components in is the intersection of a (possible empty) set of hyperplanes in .
Proof.
If has codimension , then there are hyperplanes in whose intersection is . If has codimension , then the set of hyperplanes in whose intersection is is the union of the two sets of hyperplanes that contain or , respectively. These two sets must therefore have common hyperplanes whose intersection must coincide with the span of . The last statement of the claim follows by induction. ∎
Claim 2**.**
If is irreducible, it is the scheme theoretic intersection of all hyperplanes containing . Otherwise is the scheme theoretic intersection of these hyperplanes together with the set of reducible quadrics with one component containing one of the (maximal) linear spaces that is a component in passing through while the other component contains the span of the rest .
Proof.
The first part of this assertion is clear. So we assume that is reducible. Note that is not the whole ambient space due to the simplicity of the hyperplane arrangement . Clearly is contained in the intersection of the described hyperplanes and reducible quadrics. To prove equality, let be a linear space that is a component in the intersection of these hyperplanes and the collection of reducible quadrics, one set of quadrics for each component in . Then is either one of the components in , or it is contained in the linear span of for every component . But since is simple, the intersection of all such linear spans is the intersection of all the linear components in , which proves the claim. ∎
For each , we consider the hyperplanes in that contain , and the hyperplanes in that do not contain but still pass through . Since the hyperplane arrangement is simple, we have . Furthermore, we consider the following set of Wachspress coordinates:
[TABLE]
If is reducible, we also consider a second set of Wachspress coordinates for every linear component through :
[TABLE]
To prove the theorem, it suffices to show the following two claims.
Claim 3**.**
The intersection of the hyperplanes in the zero sets of the coordinates in equals the span of .
Proof.
If spans the ambient space, and there is nothing to prove. Otherwise, for , we set
[TABLE]
Then is a linear space through that is not contained in . In fact, if it is, then it is contained in a component of that at the same time is not contained in , which is impossible. Therefore contains a face of . Since the hyperplane arrangement is simple, the linear space is spanned by vertices of . As defines a hyperplane in , these vertices cannot all lie in . Therefore at least one vertex, say , lies in . The coordinate therefore vanishes on but not on any of the other hyperplanes in that pass through , so . Notice that we have such a coordinate for each hyperplane that contains . ∎
Claim 4**.**
If is reducible, the intersection inside the span of of the reducible quadrics in the zero sets of the coordinates in equals .
Proof.
By Claim 1, both and are intersections of some of the hyperplanes , , inside the span of . So there are such that and . We set
[TABLE]
Then is a linear space through that is not contained in . In fact, if it is, then it is contained in a component of that at the same time is contained in neither nor , so is distinct from and is not contained in against the assumption that . Therefore is spanned by vertices of . None of these vertices is contained in , since any such vertex would be contained in , in particular in , which does not contain any vertices of . On the other hand, these vertices span , so they cannot all lie in . Therefore at least one vertex, say , lies in . The coordinate therefore vanishes on and on but not on any of the other hyperplanes in that pass through , so . Notice that we have such a coordinate for each hyperplane that contains but not , and each hyperplane that contains but not . ∎
Varying this construction over all decompositions of into and , we have recovered the scheme theoretic intersection of hyperplanes and reducible quadrics yielding as described in Claim 2. Moreover, we have shown that each of these hyperplanes or quadrics is contained in the zero locus of a Wachspress coordinate such that the other components in the zero locus do not pass through . This concludes the proof of the theorem. ∎
Theorems 1.10 and 1.11 imply that the Wachspress map
[TABLE]
is given by all homogeneous forms of degree which vanish along the residual arrangement . In the remainder of this section, we study the image of the Wachspress map, i.e. the Wachspress variety , as well as the image of the adjoint under . First we examine the projective space spanned by the image of the adjoint.
Lemma 3.2**.**
Let be a full-dimensional polytope in with vertices and a simple hyperplane arrangement . The dimension of is .
Proof.
Due to Proposition 1.7, Warren’s adjoint polynomial vanishes along and has degree . Here we consider its homogeneous version. Since the forms , , are linearly independent in , we can extend them to a basis of the space by adding forms, say , , , . With these coordinates we can pick a different realization of the Wachspress map:
[TABLE]
As the first forms vanish on the adjoint , we see immediately that the dimension of is at most . To prove that the dimension of cannot be smaller, it is enough to show that every hyperplane in containing is defined by a linear form in the first coordinates in (3.1). Indeed, the preimage under of a hyperplane in containing is a hypersurface in of degree which contains the adjoint , so it is the zero locus of a linear combination of the first entries in (3.1). ∎
This lemma shows that the projection from maps onto . In fact, the restriction of this projection to the Wachspress variety is the inverse of the Wachspress map .
Proof of Theorem 1.12.
From (3.1) we see that maps a point to . Hence, outside of the adjoint , this map is the identity.
A point is of the form and for some point . Again by (3.1), the map maps to . Thus, outside of , the map is the identity. ∎
In the remainder of this section we focus on polytopes in with simple plane arrangements . To show Proposition 1.15, we first exhibit the combinatorial structure of the residual arrangement and afterwards we investigate the blowup along .
Lemma 3.3**.**
Let be a full-dimensional polytope with facets and a simple plane arrangement . The residual arrangement consists of lines. Moreover, the isolated points in as well as the double and the triple intersections of the lines in satisfy
[TABLE]
Proof.
Since is simple, twice the number of its edges is three times the number of its vertices. Together with Euler’s formula we get that has vertices and edges. This shows that has lines.
Let be a set consisting of three planes in . The pairwise intersections of these planes are three lines. We denote by the number of these lines that are part of the residual arrangement . If (i.e., all three lines are edges of ), then the intersection of the three planes in is either a vertex of or an isolated point of . As has vertices, we have shown that
[TABLE]
Since a fixed line can be derived from exactly sets as the intersection of two of the planes contained in , the sum counts every line in the residual arrangement times. This yields
[TABLE]
∎
Lemma 3.4**.**
Let be a full-dimensional polytope with a simple plane arrangement . We consider the blowup along the residual arrangement .
The exceptional divisor in has one component isomorphic to over each isolated point and over each triple point, and one component isomorphic to a rational ruled surface over each line. 2. 2.
The exceptional divisor as a Cartier divisor on has multiplicity one along each component lying over isolated points and lines, and multiplicity two along each plane over a triple point. 3. 3.
The components of the exceptional divisor intersect pairwise in a line if they lie over lines through a double point, or if one component lies over a line through a triple point and the other lies over the triple point. Two components lying over lines through a triple point meet in a node in the component over the triple point. 4. 4.
The singularities of the threefold are isolated quadratic singularities, one over each double point and three over each triple point. 5. 5.
The strict transform in the blowup of a general quadric surface through the three lines of a triple point is smooth and intersects the exceptional component over the triple point in a line that does not pass through any of the nodes.
Proof.
The statements are all local over . The blowup of a smooth variety along a smooth subvariety is smooth. The exceptional divisor is a projective bundle over the subvariety, see [8, Thm 8.24]. It therefore suffices to check the statement concerning double and triple points on .
A double point on a curve is locally given by equations in . Then the blowup along the curve is defined by in , where are the homogeneous coordinates on . The exceptional set over is then isomorphic to , while the blowup is smooth except at the point where it has a quadratic singularity. The exceptional divisors over the lines and , are planes that meet along the line in the blowup.
A triple point on a curve is locally given by equations in . Then the blowup along the curve is defined by
[TABLE]
in , where are the homogeneous coordinates on . The exceptional set over is then isomorphic to . In the affine part where , the equations of the blowup are . These equations define the threefold which is smooth except at the origin , where it has a quadratic singularity. For and we get two more points, all three quadratic singularities that lie in the exceptional plane . The exceptional divisor lying over the line has two components, the plane and the plane . The two planes meet along the line . The latter plane lies over the whole line, the other only over the triple point. Similarly, over each of the two other lines there are two components, one is the common plane over the triple point, the other lies over the whole line. The three components distinct from intersect pairwise in the nodes over the triple point: when the two planes and intersect in the node .
The multiplicity of the exceptional components in the exceptional divisor as a Cartier divisor is given by the multiplicity of the generators of the ideal along the locus in that is blown up. The generators are smooth along the lines except at the triple point where the multiplicity is two.
The quadric cone passes through the three lines of a triple point. Since the blowup is defined by the locus where the matrix has rank one, the strict transform of the surface contains the line in the exceptional plane over the triple point, which does not pass through any of the nodes. ∎
In the following, we use the notation from Proposition 1.15. Let be the union of exceptional divisors of the blowup that lie over the isolated points in . Let be the union of exceptional divisors that dominate the lines in and let be the union of components lying over triple points. Thus is the exceptional divisor of the blowup as a Cartier divisor.
We use the divisor classes on to compute degrees and sectional genera of , of the image of the adjoint surface and of surfaces in linearly equivalent to the image of the plane arrangement . It would be natural to compute these degrees and genera as intersection numbers on . But is singular, so these intersection numbers are not easily defined. On the other hand, by Theorem 1.10, the singular locus of the common zeros of the Wachspress coordinates consists of the double points and the triple points of . So by Bertini’s theorem the general linear combination of Wachspress coordinates vanishes on a surface that has quadratic singularities at the triple points and is smooth elsewhere.
Quadratic singularities on are resolved by a single blowup with a -curve as exceptional divisor, a line in the exceptional plane over a triple point of according to Lemma 3.4. In particular the strict transform of is smooth on . So by restricting to we may compute the intersection numbers we need for our degree and genus computations.
We let denote the pullback to of the class of a plane in . Since has only isolated quadratic singularities, the canonical divisor on is Cartier and given by for some integer . The class of the smooth surfaces on is the pullback of hyperplane sections on to . It is given by
[TABLE]
where is the number of facets of the polytope , since has multiplicity one at the isolated points and along the lines of and multiplicity two at the triple points. By adjunction the canonical divisor on is the restriction to the surface of . The restriction of to the surface is the union of exceptional -curves lying over the quadratic singularities in . The intersection number between the canonical divisor on and these curves is zero, so we conclude that , and therefore
[TABLE]
We recall that is the linear system of polytopal surfaces of , i.e. of divisors in which have degree and vanish with multiplicity along the codimension- part of the residual arrangement. The strict transform of a general divisor is linearly equivalent to the strict transform of . Since the polytopal surface has multiplicity three along the isolated points, two along the lines in and three at the triple points, the strict transform on belongs to the class
[TABLE]
The adjoint is a surface of degree with multiplicity one along the isolated points, one along the lines, and two at the triple points, so its strict transform on belongs to the class
[TABLE]
By adjunction the restriction of to any surface in the class is the canonical divisor on this surface. The image in of any surface in the class would necessarily be a surface of degree that contains , explaining the name “adjoint” for .
We are now ready to prove Proposition 1.15. For this, we first investigate when the image of the adjoint surface is a curve.
Lemma 3.5**.**
Let be a full-dimensional polytope with a simple plane arrangement . The image of the adjoint surface under the Wachspress map is a curve if and only if is a triangular prism or a cube.
Proof.
We describe in Remark 3.1 and Example 1.13 how the Wachspress map restricts to the planes in . Each plane in is mapped birationally to a surface in and, since the vertices in are mapped to linearly independent points, distinct planes are mapped to distinct surfaces. The adjoint intersects the strict transform of in a curve in each component of . So the adjoint is mapped to a curve only if the intersection with each component of is contracted in . The intersection of the adjoint with the strict transform of a plane spanned by a facet of is trivial if the facet is a triangle. This intersection is a line contracted in if the facet is quadrangular (due to Example 1.13). If the facet has at least five vertices, the intersection with the adjoint is not contracted (again by Example 1.13). So the image of the adjoint is a curve only if every facet of has at most four vertices.
Since there are three facets through each of the vertices in (where ), we have . If every facet of has at most four vertices, this implies , which is equivalent to . There are exactly four combinatorial types of simple polytopes with at most six facets (see the first four rows of Table 1). One of them has pentagonal facets, so in that case is a surface. The adjoint of a tetrahedron is empty. By Example 1.14, the image of the adjoint of a triangular prism or a cube is a curve. ∎
Proof of Proposition 1.15.
We restrict our computations to a smooth surface in the class of on . We denote by also its restriction to . Furthermore, we denote by and the restriction to of the exceptional divisors and respectively. Clearly, and are exceptional curves on , while is the strict transform of the union of lines in . Due to (3.2), the restriction of to is
[TABLE]
The following intersection numbers on follow from Lemma 3.4:
[TABLE]
where the latter equality holds because triple points of are quadratic singularities on . Moreover, we have
[TABLE]
since there are lines in (due to Lemma 3.3). Notice that
[TABLE]
To compute the self-intersection of , we use the adjunction formula on . First we compute the arithmetic genus of as a union of lines with transverse intersections, using repeatedly the formula for the sectional genus of a reduced and reducible curve:
[TABLE]
There are lines all together in , and intersection points between these lines. So the arithmetic genus is
[TABLE]
The canonical divisor on is . By the adjunction formula, we have
[TABLE]
which yields
[TABLE]
We may now compute the degree of as the degree of the image in of :
[TABLE]
Due to (3.4) the degree of is computed as
[TABLE]
This degree is zero if is not a surface (see Table 1 and Lemma 3.5). The second expressions of the degree formulas for and found in Proposition 1.15 are obtained by plugging the relation between , and in Lemma 3.3 into the above formulas.
The sectional genus of coincides with the sectional genus of which by the adjunction formula is
[TABLE]
Similarly the genus of , when it is a surface, is
[TABLE]
For the degree and sectional genus of any surface we use the fact that these surfaces are linearly equivalent to on . Therefore the degree of equals the degree of , which is by Example 1.13. Since is the image of in , this degree may also be computed in in terms of the numbers using (3.3),(3.5), (3.6) and (3.7):
[TABLE]
The sectional genus may be computed on : The sectional genus of a component of an -gon facet is . Each line through an edge of is mapped by to a line and the simple polytope has edges. Therefore, using the formula (3.8), the sectional genus of is
[TABLE]
Again, in terms of the numbers this genus is
[TABLE]
We conclude this section with a description how the extended Wachspress map restricts to the components of the exceptional divisor described in Lemma 3.4.
Remark 3.6**.**
The restriction of the Wachspress map to the exceptional divisor on maps the components over isolated points and triple points in to planes, and the components over lines to scrolls in .
4 Polytopal hypersurfaces
In this section we consider the smooth blowup . We first prove Proposition 1.16, before we aim to classify which polytopes admit smooth strict transforms of their polytopal hypersurfaces.
Proof of Proposition 1.16.
We denote by the class of the strict transform in of the exceptional divisor over and by the class of the strict transform in of the exceptional divisor over . Let be the class of the strict transform in of a general hyperplane in . The canonical divisor on is then
[TABLE]
while the strict transform of is
[TABLE]
since has multiplicity along any component of both and . However, any polytopal hypersurface also has multiplicity along , but it may have lower multiplicity along some of the components of . Therefore the strict transform belongs to the class
[TABLE]
where each is a component of for some and . The adjunction formula on says that the class of the canonical divisor on the strict transform is the restriction to of
[TABLE]
The adjoint hypersurface has multiplicity one along each component of and multiplicity at least one along each component of , so the strict transform belongs to the class
[TABLE]
where . Thus
[TABLE]
and each is mapped by to the singular locus of . As the adjoint of the polytope is unique, by Theorem 1.5, the strict transform is the only hypersurface in whose class equals . Together with (4.1), this implies that the strict transform has a unique canonical divisor, the restriction of to . ∎
If the strict transform of a divisor is smooth, then the polytopal hypersurface has to be irreducible. We show now that many polytopes do not admit an irreducible polytopal hypersurface .
Lemma 4.1**.**
Let be a full-dimensional polytope with facets and a simple hyperplane arrangement . Moreover, let be a facet of which has facets. If or , the hyperplane spanned by is a fixed component in .
Proof.
By Corollary 2.4, the intersection is the union of hyperplanes in and , the residual arrangement of . Any hypersurface in must be singular along the hyperplanes in and have multiplicity at least two along . Therefore, if does not contain , then is a hypersurface in which contains the double hyperplanes. The residual hypersurface in has degree and is singular along . If , the facet is not a simplex and its residual arrangement is not empty. Since the adjoint hypersurface is the unique hypersurface in of degree that contains (by Theorem 1.5), any hypersurface that is singular along the non-empty residual arrangement must have degree at least . Thus, if , we have , i.e. . If , so if is a simplex, we can only conclude that , i.e. . ∎
This lemma suffices to find all polytopes in and in which have polytopal hypersurfaces with smooth strict transforms under .
Proof of Proposition 1.17.
If is a -gon, every facet of has exactly two facets. Hence, if , then Lemma 4.1 implies that the only divisor in is . Since is reducible, its strict transform cannot be smooth.
For all , the linear system has positive dimension and base points only at the points in , so by Bertini’s theorem a general polygonal curve is irreducible with simple nodes exactly at the points in . The strict transform is a smooth elliptic curve. ∎
Corollary 4.2**.**
Let be a full-dimensional polytope in with a simple plane arrangement and an irreducible polytopal surface in . The combinatorial type of is one of the nine combinatorial types in Table 1.
Proof.
Let denote the number of facets of . Since there are three facets through each of the vertices in , we have . By Lemma 4.1, since contains an irreducible element, each facet of has at least edges. Therefore , which implies . If , Lemma 4.1 further implies that cannot have triangular or quadrangular facets. Thus each of the nine facets has at least five edges, so , a contradiction. Hence, we have shown that .
All combinatorial types of simple three-dimensional polytopes with at most six facets are depicted in Table 1. Hence, we have to investigate the cases that has seven or eight facets.
Let us first assume that . There are exactly combinatorial types of simple three-
dimensional polytopes with eight facets. Two of these are depicted in the last two rows of Table 1. The other have a triangular facet. We will show now that these combinatorial types do not admit an irreducible divisor . For this, we assume for contradiction that has a triangular facet. At least one of the triangular facets of has a neighboring facet with . Due to the triangular facet, the residual arrangement has an isolated point , which is contained in the plane spanned by the facet . By Corollary 2.4, the intersection consists of lines and the residual arrangement of the facet which is points, among them . An irreducible polytopal surface intersects in a curve of degree eight, which is double along the lines. The residual curve in has degree and multiplicity at least two at the points of . At the isolated point , the curve must even have multiplicity three. We observe that (since ). If , there is a unique curve of degree with multiplicity two at the points (namely the adjoint doubled), but this curve does not have multiplicity three at . If , there is a pencil of curves of degree with multiplicity two at the nine points of , but none with the additional multiplicity three at . This contradicts the construction of the curve .
Let us now assume that . There are exactly five combinatorial types of simple three-dimensional polytopes with seven facets. Three of them are shown in Table 1. We will show in the following that the remaining two combinatorial types do not admit an irreducible divisor . Both combinatorial types are depicted in Figure 3. In both cases there is a hexagonal facet . The plane of this facet contains three isolated points of that are not collinear. In fact the plane intersects in exactly six points on lines in in addition to the isolated points. Through the nine points there is an irreducible plane cubic curve . Any irreducible polytopal surface would intersect the plane in a curve of degree seven with double points at and triple points at . We consider the intersection on with and with the union of the six lines spanned by the edges of the hexagon . The latter is a curve with a node at each of the nine points, so on the intersection with forms a divisor of degree that is a complete intersection. Unless is a component of , their interection forms a divisor of degree that is also a complete intersection. The three isolated points form a divisor on that is precisely the difference between these two divisors. It is therefore also a complete intersection, i.e. the intersection of with a line, which contradicts the observation that are not collinear. If is a component of , then we may use a similar argument for , using the fact that do not lie on a conic. These contradictions imply that the plane must be a component of any polytopal surface. ∎
Remark 4.3**.**
We verified with symbolic computations in Macaulay2 [7] that, for each combinatorial type in Table 1, the general polytopal surface of a general polytope of type is irreducible and its strict transform in is smooth.
Corollary 4.2 and Remark 4.3 show that the combinatorial types of three-dimensional simple polytopes which admit smooth strict transforms of polytopal surfaces are exactly the combinatorial types listed in Table 1. Our next task is to determine the birational type of the general polytopal surfaces associated to these combinatoral types. This is immediate for tetrahedra. For a general polytope of one of the other combinatorial types, we determine the canonical curves on its polytopal surfaces, starting with the canonical curve on the strict transform of .
Example 4.4**.**
When is a tetrahedron, its residual arrangement is empty. Any quartic surface in belongs to , so .
Lemma 4.5**.**
*Let be a full-dimensional polytope with facets and a simple plane arrangement . On , the canonical curve consists of exceptional curves and strict transforms of curves in the planes of . The former are all contracted by the Wachspress map, while the latter are the strict transforms of the unique adjoint curve of the polygonal facet in each plane. These adjoint curves have degree in the plane of an -gon, while their images by the Wachspress map are pairwise disjoint and have degree respectively. In particular, the degree of the image of by the Wachspress map is *
Proof.
The blowup resolves the base locus of the Wachspress map . Thus the exceptional locus of the morphism is contracted by the Wachspress morphism . So for the lemma it suffices to consider the strict transform of and its components in .
We consider the strict transform of a plane in spanned by an -gon facet . Let be the restriction of the exceptional divisor over an isolated point and let be the restriction of the exceptional divisor that dominates a line . We denote by the pullback to of the class of a plane in . We observe that is transverse to the strict transform of the other planes along the strict transform of the union of the lines spanned by the edges of the -gon . The curve belongs to the class restricted to . The class of the canonical divisor on is . The restriction of the canonical divisor on to is equivalent to the sum of the canonical divisor on and the intersection of with the residual . So
[TABLE]
which is the class of the strict transform of the unique adjoint curve of the -gon . Thus, the canonical curve on is the union of the strict transforms of the adjoint curves of the facets of . Due to Remark 3.1 and Example 1.13, the degree of the image of the canonical curve by the Wachspress map is
It is enough to show that the strict transforms of the adjoint curves are pairwise disjoint. For this, we consider two facets and of with and vertices, respectively. The planes and spanned by these facets intersect in a line .
We assume first that the line contains an edge of . In that case, the adjoint intersects precisely in the points of intersection on with the other lines in which are not spanned by the two neighboring edges in of the edge spanning . The analogous statement holds for . Hence, the adjoint curves and intersect exactly at the isolated points of which are contained in . These points are blown up on . Since and intersect transversally along , the adjoints and meet transversally along their points of intersection, so their strict transforms do not intersect on .
If does not contain any edge of , then is a line in the residual arrangement and is blown up on . Since and meet transversally along , their strict transforms do not intersect on . Therefore the two curves and are also disjoint. ∎
Corollary 4.6**.**
Let be a full-dimensional polytope with a simple plane arrangement and a smooth strict transform in of a polytopal surface . If has a hexagonal facet, then the general such is an elliptic surface. If has only triangular, quadrangular and pentagonal facets, then general such is a -surface.
Proof.
We first show that is regular, i.e. . Since , we deduce from the cohomology of the exact sequence
[TABLE]
that . But by Theorem 2.1, so is regular.
Next, the smooth surface has a unique canonical curve, so . If has no hexagonal facets, then, by Lemma 4.5 and Theorem 1.18, the canonical curve is the union of exceptional rational curves contracted by the Wachspress map and rational curves mapped to lines, hence so is also . These curves must be exceptional curves of the first kind on , i.e. a minimal model of is a -surface. If has a hexagonal facet, then, by Lemma 4.5, contains in addition plane cubic curves, hence so does also the canonical curve on . In particular, a minimal model of is a regular elliptic surface. ∎
Proof of Theorem 1.18.
If the strict transform in of a polytopal surface is smooth, then is, of course, irreducible. In that case, Corollary 4.2 implies that the combinatorial type of is listed in Table 1.
By Remark 4.3 every combinatorial type in Table 1 admits a smooth strict transform in of a polytopal surface . Their birational types are determined by Corollary 4.6. ∎
In the remainder of this article, we provide more detailed descriptions of the Wachspress maps and polytopal surfaces associated to general polytopes of the combinatorial types listed in Table 1. We use the notation from Proposition 1.15 in what follows.
4.1 Prisms
Example 4.7**.**
In Example 1.14 we already described , and for triangular prisms and perturbed cubes .
When is a triangular prism, then , in the Segre embedding and the image of the adjoint plane is a line . Let be a general polytopal surface for . Then is a -surface of degree and genus with Picard rank : it is the intersection of the Segre threefold with a quadric hypersurface. The surface is the projection of from three collinear points, namely the intersection of with the line .
When is a cube (), then and in the Segre embedding. For a general polytopal surface , the image is a -surface of degree and genus : it is the intersection of with a quadric hypersurface. The image of the adjoint quadric surface is a twisted cubic curve that intersects in points. The surface is the projection of from these points.
From here on we assume that . Any two planes of quadrangular facets in the prism that do not share an edge, intersect in a line that belongs to the residual arrangment . In fact there are such lines. The two planes of -gon facets of the prism intersect in the remaining line in , that we denote by . There are no isolated points in and the line is disjoint from the other lines in . The double and triple points of therefore all lie on the intersections of three planes of quadrangular facets. If exactly two of the facets share an edge in , there are two lines in through the point, and if no two of the three facets share an edge, there are three lines through the point that span . The number of double points in is therefore , while the number of triple points is . In there is a pencil of surfaces: the strict transforms of planes through the line . The planes of the quadrangular facets of cut each plane through in a -gon. So the restriction of to any of the surfaces is the Wachspress map with respect to that polygon. By Example 1.13, the image of such a surface has degree and spans a , which means that lies in a rational normal scroll in , where . In particular the degree of this scroll is one more than its codimension, i.e. , and its ideal is generated by quadrics.
Example 4.8**.**
When is a pentagonal prism (; see row in Table 1), then computations with Macaulay2 show that dim and that the ideal of is arithmetically Gorenstein, generated by quadrics. is a rational quartic scroll in a . Indeed, the -fold lies in a -scroll over . Each intersects in a surface , in this case a quartic Del Pezzo surface. There are no isolated points and no triple points in , so there is an equivalence on (where is the pullback of the class of a hyperplane section on to ), and the general surface equivalent to is a smooth quartic surface containing , the union of a -cycle of lines and a disjoint line . The -fold therefore has at most isolated singularities: it is a -fold of degree and any smooth curve section is a tetragonal canonical curve. The adjoint surface is a cubic surface through that is mapped to in a . The plane sections that contain are mapped to lines in , so is a rational normal quartic scroll. Moreover, the image of a general polytopal surface is a non-minimal -surface with two -lines of degree and genus . The surface is the projection of from , a that contains the two exceptional lines and intersects in five more points.
Example 4.9**.**
When is a hexagonal prism (; see row in Table 1), then computations with Macaulay2 show that dim and that the ideal of is generated by quadrics and cubics. Residual to in the quadrics is a rational -scroll of degree that intersects in the image of the adjoint surface, a -surface of degree and genus . Indeed, the -fold lies in a -scroll over . Each of this scroll intersects in the image of a surface , in this case a surface of degree and genus . The adjoint surface is a quartic surface, and in a is a -surface of degree and genus . The plane sections that contain are mapped to plane cubic curves in . So lies in a rational normal -fold scroll of degree in a and this scroll is contained in every quadric that contains . Moreover, the image of a general polytopal surface is a minimal elliptic surface of degree and genus . The surface is the projection of from , a that contains two plane cubic curves and intersects in six more points.
4.2 Non-prisms
Example 4.10**.**
When is a non-cube simple polytope with six facets (see row in Table 1), then computations with Macaulay2 show that dim and that the ideal of is generated by quadrics. The residual arrangement consists of three lines where and are skew lines that both intersect , and two isolated points that span a line that does not intersect any of the other three lines . The adjoint surface is a smooth quadric surface that contains . It is mapped by to a smooth quadric surface in a . The planes through are mapped to Del Pezzo surfaces of degree in . Each one of them is a complete intersection of two quadrics in a , so is contained in a five-dimensional rational normal -scroll over of degree in . Moreover, the image of a general polytopal surface is a non-minimal -surface with two -lines of degree and genus . The surface is the projection of from , a that is spanned by the exceptional lines and intersects in two more points.
Example 4.11**.**
When is a simple polytope with seven facets, including two hexagonal facets (see row in Table 1), then computations with Macaulay2 show that dim and that the ideal of is generated by quadrics and cubics. Residual to in the quadrics is a Segre threefold, , that intersects in the surface , a rational surface of degree and genus . The residual arrangement consists of six lines and three collinear isolated points. Let be the line containing the three isolated points. The Wachspress map maps the line to a line in . The planes through generate a pencil. The residual arrangement intersects each of these planes in points. These planes are mapped by to rational surfaces of degree and genus that each span a . Any two of these surfaces span . Therefore is contained in a rational normal -scroll over , which has degree and is a cone with vertex over a . The unique plane cubic curve through the nine points is the intersection of each of these planes with the adjoint surface . These plane cubic curves are mapped by to plane cubic curves in . Moreover, the image of a general polytopal surface is a minimal elliptic surface of degree and genus . The surface is the projection of from , a spanned by two plane cubic curves that intersects in three more points.
Example 4.12**.**
When is obtained by cutting off a vertex of a perturbed cube (see row in Table 1), then computations with Macaulay2 show that dim and the ideal of is generated by quadrics. The adjoint surface is a smooth cubic surface that contains . It is mapped by to a Del Pezzo surface of degree in a . Moreover, the image of a general polytopal surface is a non-minimal -surface with three -lines of degree and genus . The surface is the projection of from , a that is spanned by the three exceptional lines and intersects in three more points.
Example 4.13**.**
When is a non-prism simple polytope with eight facets, none of which are triangles (see last row of Table 1), then computations with Macaulay2 show that dim and that the ideal of is generated by quadrics. The residual arrangement consists of ten lines with no triple points that form a curve of arithmetic genus . There are two lines in that are intersections of two planes of pentagons. Let be one of these lines. Then intersects two other lines in , so a plane through is intersected by seven lines of outside . This plane is mapped by to a surface of degree and genus in a in . The images of any two planes in the same pencil span the whole , so is contained in two rational normal -scrolls over , each of codimension , so of degree . For each line in of intersection between a plane of a quadrangle and plane of a pentagon there is a similar associated -scrolls over that contains , each of codimension , so of degree . The adjoint surface is a smooth quartic surface that contains . It is mapped by to a surface of degree and genus in a . Moreover, the image of a general polytopal surface is a non-minimal -surface with four -lines of degree and genus . The surface is the projection of from , a that is spanned by the four exceptional lines and intersects in four more points.
Acknowledgements.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Fall 2018 semester. We are grateful to Rainer Sinn, Frank Sottile, Hal Schenck and Dmitrii Pasechnik for many insightful discussions. We also thank the anonymous referees for helpful suggestions that improved our exposition.
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