How to repair tropicalizations of plane curves using modifications
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Our approach is based on the close connection between analytic curves (in the sense of Berkovich) and tropical curves. We investigate the effect of these tropical modifications on the tropicalization map defined on the analytification of the given curve.
Our study is motivated by the case of plane elliptic cubics, where good embeddings are characterized in terms of the -invariant. Given a plane elliptic cubic whose tropicalization contains a cycle, we present an effective algorithm, based on non-Archimedean methods, to linearly re-embed the curve in dimension 4 so that its tropicalization reflects the -invariant. We give an alternative elementary proof of this result by interpreting the initial terms of the -discriminant of the defining equation as a local discriminant in the Newton subdivision.
Key words and phrases:Tropical geometry, tropical modifications, Berkovich spaces, elliptic curves, discriminants
2010 Mathematics Subject Classification:14T05, 51M20, 14H52, 14G22
Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants. Often, we can derive statements about algebraic varieties by means of these (easier) combinatorial objects. One general difficulty in this approach is that the tropicalization strongly depends on the embedding of the algebraic variety. Thus, the task of finding a suitable embedding or repairing a given “bad” embedding to obtain a nicer tropicalization becomes essential for many applications. The purpose of this paper is to advertise tropical modifications as a tool to locally repair embeddings of plane curves, as suggested by Mikhalkin in his ICM 2006 lecture .
An important and motivating example is the case of elliptic curves. In [17, Example 3.15], Mikhalkin proposed the cycle length of a tropical plane elliptic cubic to be the tropical counterpart of the classical -invariant. Inspired by this remark and using Gröbner fan techniques, Katz, Markwig and the second author proved that when the elliptic cubic is defined over the Puiseux series field, the valuation of the -invariant is generically reflected on the cycle length of the tropical curve . For special choices of coefficients, this length can be shorter than expected. These non-generic situations have a very explicit characterization. First, the cycle in the tropical curve must contain a vertex of valency at least four, and second, the initial form of the discriminant of the cubic must vanish. Thus, in the case of plane elliptic cubics, or more generally, for elliptic curves embedded smoothly into a toric surface, the question of what constitutes a good embedding from the tropical perspective has a precise answer: the cycle length should reflect the negative valuation of the -invariant.
One of the main contributions of the present paper is an algorithm that recursively repairs bad embeddings when the tropical plane elliptic cubic contains a cycle. The power of Algorithm 4.8 lies in its simplicity: it only uses linear tropical modifications of the plane, and linear re-embeddings of the original curve. Furthermore, this result is achieved in dimension 4. This approach has an additional advantage. Rather than drastically changing the polyhedral structure of the input tropical curve, it keeps its relevant features. It only adds missing edges and changes tropical multiplicities. The output tropical curve has the expected cycle length. We view this as a possibility to “locally repair” the problematic initial embedding.
The case of elliptic curves suggests itself as a playground for uncovering the deep connections between Berkovich’s theory, tropicalizations, and re-embeddings. More concretely, let be a smooth elliptic curve and be a semistable regular model of over a discrete valuation ring. Let us assume that has bad reduction. Then, the minimal Berkovich skeleton of the complete analytic curve is homotopic to a circle and it can be obtained from the dual graph of the special fiber of . Foundational work of Baker, Payne and Rabinoff proves that when the embedding induces a faithful tropicalization on the cycle, the length appearing in the minimal Berkovich skeleton induced by its canonical metric equals the corresponding lattice length in the tropicalization of [3, Section 6]. Notably, [3, Section 7] provides examples where the cycle in a tropicalization of a smoothly embedded elliptic curve is shorter, or longer, than the negative valuation of the -invariant. Good embeddings of elliptic curves with bad reduction are those where the minimal skeleton of the complete analytic curve is reflected in the associated tropical curve.
Characterizing good embeddings of curves in terms of their minimal Berkovich skeleta has one clear advantage compared to the study of tropicalizations: it is intrinsic to the curve. Work of Payne shows that the Berkovich space is the limit of all tropicalizations of with respect to closed embeddings into quasiprojective toric varieties (see [18, Theorem 4.2]). We view as a topological object incorporating all choices of embeddings.
After the investigation by Baker, Payne and Rabinoff , the meaning of suitable embeddings of curves for tropicalization purposes becomes precise: they should induce faithful tropicalizations. That is, the corresponding tropical curve must be realized as a closed subset of , and this identification should preserve both metric structures. Faithful tropicalizations of Mumfurd curves of genus 2 have been recently studied by Wagner in . In the case of plane elliptic cubics, we can reinterpret the main result of  in the language of Berkovich’s theory by saying that the tropicalization to a -valent cubic is always faithful on the cycle. This follows from the fact that all edges in a tropical cubic with a cycle have multiplicity , see [3, Theorem 6.24 and 6.25].
Two natural questions arise from the previous discussion. First, can we effectively construct embeddings of a given curve that induce faithful tropicalizations? Can we do so without computing a minimal Berkovich skeleton of the complete curve? Again, the case of elliptic cubics is a fantastic playground for exploring this question, since the faithfulness on its cycle can easily be characterized in terms of the -invariant. Following this approach, Chan and Sturmfels described a procedure to put any given plane elliptic cubic with bad reduction into honeycomb form . The honeycomb form is -valent and has edges of multiplicity , hence it induces a faithful tropicalization and the cycle has the expected length. Although running in exact arithmetic, their method involves the resolution of a univariate degree 6 equation. Each solution is expressed as a Laurent series in the sixth root of the multiplicative inverse of the -invariant. The solution is constructed recursively, one term at a time. This re-embedding completely alters the structure of the original tropical curve.
In contrast, our approach allows us to give a positive and effective answer to the questions above. Our algorithm to repair embeddings of plane elliptic cubics relies on methods we develop in Section 3 for arbitrary plane curves. Theorems 3.2 and 3.4 allow us to locally repair certain embeddings of curves using a linear tropical modification of the plane. They should be viewed as a partial answer to the questions above. They hold under certain constraints imposed by the local topology of the input tropical curve. Nonetheless, these two technical results suffice to completely handle the case of plane elliptic cubics. As a byproduct, we enrich Payne’s result [18, Theorem 4.2] for plane elliptic cubics connecting the Berkovich space to the limit of all tropicalizations by a concrete procedure that gives the desired tropically faithful embedding using only linear tropical modifications of the plane. Our experiments in Section 5 suggest that the techniques introduced in this paper may be extended to other combinatorial types of tropical curves, although new ideas will be required to generalize Theorems 3.2 and 3.4.
We have mentioned already that -discriminants of cubic polynomials play a key role when studying the -invariant of a plane elliptic cubic. In the same spirit, Theorems 3.2 and 3.4 also involve “local discriminants” associated to certain maximal cells in the Newton subdivision of the input plane curve. In Section 4.3 we derive Algorithm 4.8 in an elementary fashion, by relating the global discriminant of the cubic to the local discriminants mentioned above. This is the content of Corollary 4.18. Theorem 4.15 provides a factorization formula for initial forms of discriminants of planar configurations. We expect this result to have further applications besides Algorithm 4.8.
The rest of the paper is organized as follows. In Section 2.1, we introduce notation and discuss background on tropicalizations, modifications and linear re-embeddings. In Lemma 2.2, we characterize linear re-embeddings of plane curve induced by tropical modifications along straight lines in terms of charts and coordinates changes of . Thus, we can visualize the repaired embeddings by means of collections of tropical plane curves. In Sections 2.2 and 2.3 we discuss preliminaries involving Berkovich skeleta and -discriminants, which play a prominent role in our study.
In Section 3, we present our two main technical tools to locally repair embeddings of smooth plane curves by linear re-embeddings. Our proof builds upon Berkovich’s theory, -discriminants of plane configurations, and Lemma 2.2. By using linear tropical modifications and coordinate changes of , the tropical re-embedded curve will faithfully represent a subgraph of a skeleton of the analytic curve induced by its set of punctures.
In Section 4, we focus our attention on plane elliptic curves and present Algorithm 4.8. We provide two independent proofs of its correctness. The first one relies on the techniques developed in Section 3 and is discussed in Section 4.2. The second one is elementary: it is based purely on discriminants of plane configurations. We present it in Section 4.3. The main result of this section is Theorem 4.15, which relates global and local discriminants of planar point configurations.
2. Preliminaries and Motivation
We work with ideals defining irreducible subvarieties of a torus and denote their tropicalizations by (see e.g. ). Throughout this paper we often work with complete curves and their minimal Berkovich skeleta. For this reason, we always consider our ideals inside honest polynomial rings rather than Laurent polynomial rings. For tropicalization, we consider the intersection of the curve with the algebraic torus in the given embedding.
For concrete computations, we fix the field of generalized Puiseux-series , with valuation taking a series to its leading exponent. For tropicalizations, we use the negative of the valuation, i.e. the max-convention. We denote our algebraic coordinates by , whereas we indicate the tropical coordinates by . We use analogous conventions in higher dimensions.
2.1. Tropical modifications and linear re-embeddings
Let be a linear tropical polynomial with and in . The graph of considered as a function on consists of at most three linear pieces. At each break line, we attach two-dimensional cells spanned in addition by the vector (see e.g. [1, Construction 3.3]). We assign multiplicity to each cell and obtain a balanced fan in . It is called the modification of along .
Let be a lift of , i.e. , and . We fix an irreducible polynomial defining a curve in the torus . The tropicalization of is a tropical curve in the modification of along . We call it the linear re-embedding of the tropical curve with respect to .
For almost all lifts , the linear re-embedding coincides with the modification of along , i.e. we only bend so that it fits on the graph of and attach some downward ends. However, for some choices of lifts , the part of in the cells of the modification attached to the graph of contains more attractive features. We are most interested in these special linear re-embeddings. The following example illustrates this phenomenon.
We fix a plane elliptic cubic defined by
We aim to modify the tropical curve along the vertical line
in . This line corresponds to a tropical polynomial
. Its lifting is of the form where
has valuation 0. The tropicalization
depends only on the initial coefficient of . Indeed, unless this
coefficient is one, this tropical curve coincides with the
modification of along .
Figure 1 shows the special linear
re-embedding when .
Our main focus in Section 3 will be on modifications of along vertical lines. These modifications are induced by tropical polynomials of the form , with . Their liftings are of the form where has valuation 0. As we see in Figure 1, the modified plane contains three maximal cells:
By construction, is the unique cell of the modification of attached to the graph of . We let denote the relative interior of the cell , for .
We describe by means of two projections:
the projection to the coordinates produces the original curve ,
the projection gives a new tropical plane curve inside the cells and , where . The polynomial generates the elimination ideal .
Notice that the projection gives no information about since it maps any tropical curve to the tropical line with vertex . The following lemma explains how to reconstruct the curve inside this modified plane using the two relevant projections above.
The linear re-embedding in the modification of along the linear tropical polynomial is completely determined by the two tropical plane curves and , where . In particular, the vertices of along the line are the endpoints of the connected components of .
First, we fix a point with either or , thus in the relative interior of one of the cells , for . We claim that belongs to if and only if one of the following two conditions hold:
The first implication follows directly from the Fundamental theorem of tropical algebraic geometry (see e.g. [16, Theorem 3.2.5]) and the fact that . For the converse, we use the same result to lift a point with to a unique point in . This point satisfies . Analogously, any point lifts uniquely to a point in , where .
It follows that the set of points in outside the line is completely determined by the two projections and . It remains to prove that we can also detect the tropical multiplicities and all points in from these two projections. To see this, notice first that the multiplicities of all edges of whose relative interior lies in for coincide with the corresponding multiplicities of the projected edges in or , respectively. This follows from the unique lifting property discussed above and the generalized push-forward formula for multiplicities of Sturmfels-Tevelev in the non-constant coefficients case [3, Theorem 8.4]. We can also compute the multiplicity of an edge on the line by comparing the multiplicities of the preimages of the edge in the two charts. An edge of multiplicity zero should be interpreted as a phantom edge. This concludes our proof. ∎
Using the previous result we can visualize the modification of along a vertical line and the effect of the linear re-embedding on the tropical curve by means of the two relevant projections. The colors and cell labels on the projections and the modified plane in Figure 1 indicate the nature of the fibers of each projection. The dashed line on each projection represents the image of the vertical line used to modify . We keep these conventions throughout this paper.
By Lemma 2.2 we know that the features of are encoded in the polynomial . For special values of , the Newton subdivision of is unexpected and yields an interesting behavior in . We observe this phenomenon in Figure 1: the cycle on the tropical curve was placed to the right of the vertical line , but in this cycle has been prolonged and its leftmost vertical edge has been pushed from the line to the line . This example illustrates the general principle described in the title of this paper. We discuss it further in Section 3.
As we mention earlier, our goal is to use linear tropical modifications to repair embeddings of plane curves. Let be a linear ideal defining a plane in . We re-embed the curve via the ideal . As in Lemma 2.2, we can construct from suitable 2-dimensional projections.
In order to do so, we find generators of adapted to a fixed 2-cell of . We let be the local coordinates of . Then, the corresponding variables must be linearly independent on and we can find unique polynomials for such that
Le and fix a two-dimensional cell of with local coordinates containing . Then, the ideal is isomorphic to the localization , where and is the the multiplicatively closed set generated by all , .
To simplify notation, we consider all initial ideals in the statement defined by , rather than by the projection . By definition, .
For each we write for suitable . In order to prove the statement, we study the interplay of with . By (2.1), any point in the plane defined by with is uniquely determined by its coordinates. Since , the fundamental theorem of tropical algebraic geometry ensures that
Hence, and we conclude that because . Therefore, the generators from (2.1) give a basis to compute , i.e. , In particular, all elements of are units in .
As a consequence, we construct an isomorphism by
This map induces an isomorphism between the ideals and . To prove the statement, we show that generates the quotient ideal .
Recall that and the elements of are units in the domain of . We pick a -homogeneous polynomial and show that . By [11, Lemma 2.12], we know that is the initial form of an element . We write . Since contains in its support but , an easy induction on ensures that . Thus, , as we wanted to show. This concludes our proof. ∎
2.2. Berkovich skeleta of curves and faithful tropicalization
In this section, we outline the required background on Berkovich analytic curves, their skeleta and their relationship with tropicalizations of curves. For the sake of brevity and simplicity, we restrict our exposition to the topological aspects of analytic curves. These features are captured by skeleta of curves. We follow the approach developed by Baker, Payne and Rabinoff in [3, 4].
Let be an algebraically closed, complete non-Archimedean valued field with absolute value . Our main example of interest is , i.e. the field of generalized Puiseux series. Given an algebraic curve defined over we let denote its analytification. The analytification of an affine curve is the space of multiplicative seminorms that satisfy the non-Archimedean triangle inequality and extend the absolute value on . Its topology is the coarsest one such that all evaluation maps are continuous for . The analytification of a general curve is glued from the analytification of an affine open cover. It can be shown that possesses a piecewise linear structure and it is locally modeled on an -tree [4, ]. The -points of are embedded as a subset of the leaves of this tree. The complement of the set of leaves carries a canonical metric given by shortest paths.
In , Berkovich introduced the notion of skeleta of an analytic space as suitable polyhedral subsets that capture the topology of the whole space. They are constructed from semistable formal models. Equivalently, they can be defined by means of semistable vertex sets of [4, , Theorem 1.3]. They have the structure of a finite metric graph with vertex set . We denote them by . For any choice of , there exists a deformation retract
We say is a minimal skeleton of if is minimal.
Such minimal skeletons exist by [4, ]. The Stable reduction theorem ensures that if the Euler characteristic of is at most 0, then there is a unique set-theoretic minimal skeleton of [4, Theorem 4.22]. This is the case when is smooth and non-rational. In this situation, we write , or whenever is defined by the ideal .
From now on, let us assume that is a smooth connected algebraic curve over and let denote its smooth completion. Let be its set of punctures. These punctures are contained in distinct connected components of . By construction, a semistable vertex set of is also a semistable vertex set of . In particular, by [4, Proposition 3.13] we know that . The closure of in equals . We call it the extended skeleton of with respect to and the punctures and we denote it by . Whenever the minimal skeleton of is unique, as in Example 2.5 below, the extended skeleton depends solely on the set of punctures. Following the previous notation, when the smooth, non-rational curve is defined by an ideal , we write for the complete extended skeleton.
Example 2.5 (Elliptic curves).
Let be a smooth elliptic curve defined over . If
has good reduction, then the minimal skeleton of
is a point. If has bad reduction,
then the minimal skeleton of is homeomorphic to a
circle: its corresponding semistable vertex set is a point [3, ]. Larger
semistable vertex sets will yield larger skeleta obtained from
by attaching finite trees to this circle along
points in .
From the previous discussion, it is clear that skeleta of analytic curves share many properties with tropicalizations of algebraic curves. Their interplay was studied in depth by Baker, Payne and Rabinoff in . As we next discuss, the precise relationship is captured by the tropicalization map and Thuillier’s non-Archimedean Poincaré-Lelong formula [4, Theorem 5.15].
Let be an embedded curve, and fix a basis of the character lattice of the torus. Let be the image of for . The tropicalization map given by extends naturally to a continuous map
The image of this map is precisely the tropical curve [13, ]. Given any semistable vertex set of , the map (2.5) factors through the retraction by [4, Theorem 5.15 (1)]. In particular, the resulting map
is a surjection. This last map will be our main focus of interest.
By the Poincaré-Lelong formula [4, Theorem 5.15], the maps from (2.5) and (2.6) are piecewise affine, with integer slopes. Furthermore, they are affine on each edge of the skeleton . The stretching factor on each edge is known as its relative multiplicity. If an edge gets contracted to a single point in , we set . The map is harmonic, i.e. the image of every point in and under is balanced in the following sense: only finitely many edges in the star of a point in (resp. ) are not contracted by , and these edges satisfy the identity
Here, denote the tangent directions of , i.e. the nontrivial geodesic segments starting at , up to equivalence at (as in [4, ]). The outgoing slope is if contracts and it equals times the primitive direction of the edge of that contains the (possibly unbounded) segment .
By refining the polyhedral structure of we may assume that the map from (2.6) is a morphism of 1-dimensional complexes. The balancing condition yields the following identity between tropical and relative multiplicities, as in [3, Proposition 4.24]:
By [3, Proposition 4.24], this formula can also be used to relate tropical and relative multiplicities of vertices on tropical curves and vertices of skeleta of analytic curves, when the map from (2.6) is a morphism of 1-dimensional complexes. As in the case of edges, the tropical multiplicity of a vertex of counts the number of irreducible components (with multiplicities) in the initial degenerations of the input ideal defining with respect . Rather than giving the precise definition for the relative multiplicity of a vertex in , we present two of its crucial properties, as in [3, Corollary 6.12]. Namely, is a non-negative integer and if and only if belongs to an edge of mapping homeomorphically onto its image via .
Consider a skeleton of and a finite subgraph on it. We say a closed embedding faithfully represents if maps homeomorphically and isometrically onto its image in .
Using embeddings of curves in proper toric varieties that meet the dense torus, we can extend the previous definition to complete curves. We consider those toric varieties for which the morphism is a closed immersion and use the extended tropicalization maps from , obtained by gluing the previous constructions on each toric strata along open inclusions, with the convention that .
We say that is faithful if it faithfully represents a skeleton of . By definition, a faithful tropicalization of restricts to a homeomorphism from a suitable skeleton of to a subgraph of the tropical curve . Thus, constructing an embedding of the given curve that yields such a homeomorphism can be viewed as a first step towards a faithful tropicalization of curves. Relative multiplicities on edges and the isometric requirements should be address in a second step.
In Section 4, we focus our attention on tropical faithfulness of plane elliptic cubics with bad reduction, embedded in or in a surface in . Their completions admit a closed embedding . The minimal skeleton of lies in and is homeomorphic to a circle. Our goal is to find a linear re-embedding of a given curve that faithfully represents .
We first discuss how to detect non-closed embeddings of skeleta by looking at the tropical curve.
Let be a tropicalization of the plane curve defined by , and a vertex of . We say that is locally reducible if the star of in is a reducible -dimensional complex that is balanced at , i.e. if it can be written as the union of two non-zero complexes with multiplicities that are balanced . In particular, if is the union of edges adjacent to with multiplicities , we can find with for all such that the resulting complex with multiplicities is balanced at , contains an edge of positive multiplicity and it does not agree with as complexes with multiplicities.
Consider a non-rational smooth curve defined by an ideal and let be extended skeleton defined with respect to the set of punctures . Assume that is not a closed embedding. Then one of the following conditions hold:
has an edge of higher multiplicity, or a locally reducible vertex with ;
faithfully represents a unique subgraph of .
To simplify notation, write . After refining the structure of and , we may assume without loss of generality that has no loop edges and that is a map of connected 1-dimensional abstract complexes.
Since is not a closed embedding, one of the following conditions hold:
the images of several edges intersect in more than a point;
there exists a vertex of where the fiber in is not connected;
there exists a vertex of such that is connected and it is not a singleton.
First, assume that (i) holds and let be a segment of an edge of where the images of several edges overlap. We conclude from (2.8) that lies in an edge of of higher multiplicity.
We now analyze conditions (ii) and (iii). We consider the stars of all vertices in the abstract cell complex . By (2.7) we know that the images of all stars under the tropicalization map are balanced at . In particular,
The decomposition in the right-hand side of (2.9) contains at least one non-singleton component. In order to show that is a locally reducible vertex, we seek to find two vertices where and are both nontrivial. In this situation, [3, Proposition 4.24, Corollary 6.12] ensure that .
To simplify notation, fix . Assume (ii) holds, and decompose into its connected components , where . Each component is closed in . Since is connected, we conclude that for all . Since is a morphism of complexes, we can pick a vertex in for each . By construction, for all . We conclude that is locally reducible and .
Finally, assume (iii) holds. Then the fiber in is a connected graph with at least two vertices. If is not locally reducible, then the decomposition (2.9) is trivial, and so there is a unique vertex of whose star in does not map entirely to under . We conclude that and is connected and surjects onto via . We conclude that each edge in is the image of at least one edge in . Thus, we can construct a subgraph in that is homeomorphic to by . In addition, assuming condition (i) does not occur, we know that and induces an isometry between and .
As a consequence, if condition (1) in the statement fails, by iterating the previous construction over all vertices of , we can find a unique subgraph of that maps isometrically to under the map . This concludes our proof. ∎
From the proof of Lemma 2.8 we can also extract the following information. Assume that the images under of two adjacent edges and of with a unique common endpoint are two line segments that partially overlap. Call and the non-common endpoints of and , and assume . Then, the point will be a locally reducible vertex of and its star contains the straight line with direction . In the case of complete overlap, the vertex will also be locally reducible. We know that contains the high multiplicity edge , but we cannot guarantee that it contains a straight line. Finally, when the edges and of have two common endpoints and do not get contracted by , their image will be contained in an edge of of multiplicity .
Let be an elliptic cubic curve over with bad reduction, embedded linearly by an ideal . Assume contains a cycle but is not faithful on the cycle. Then, the cycle contains a locally reducible vertex and .
We write , and assume that is a morphism of 1-dimensional complexes. Since has bad reduction, we know that contains a unique cycle . The tropical cycle is contained in , but the latter may also contain other edges of .
Next, we analyze . Since is defined by a cubic polynomial in the plane, all the edges in the cycle of have multiplicity 1. Given an edge of the cycle of , expression (2.8) ensures that exactly one edge of lies in and, moreover, this edge lies in and induces an isometry between and . Since is not faithful on the cycle of , we know that contains at least one edge that either gets contracted by or that map to an edge of outside the cycle. In both cases, we can find two distinct vertices of that map to the same vertex in the cycle of and are contained in two edges of that are mapped isometrically to edges in the cycle of . By [3, Corollary 6.12], . The decomposition (2.9) ensures that is locally reducible, as desired.
For the reverse inequality, we analyze the combinatorics of the support of , i.e. of the dual cell to in the Newton subdivision of . By Figure 10, this support is a trapezoid of height 1 and one of whose basis has length 1. Therefore, has at most two components, i.e. . This concludes our proof. ∎
Consider a smooth non-rational plane curve in defined by an irreducible polynomial and its linear re-embedding via the ideal as in Section 2.1. This re-embedding alters the skeleton of the analytic curve in a concrete way. Consider completions of these two curves, their sets of punctures and and the corresponding extended skeleta and . Notice that . These skeleta only differ by some additional ends that we attach to to obtain (see Figure 1). The bounded part of can be identified with the corresponding bounded part of using the following commutative diagram:
This diagram allows us to define two key notions: decontraction and unfolding of edges via linear re-embeddings.
If contracts a fixed bounded edge but does not, we say that the linear re-embedding decontracts this edge.
Assume next that a segment in a bounded edge of is obtained by overlapping the images of several edges of in more than one point. Refine the structure of and let be this segment. If is the union of images of finitely many edges from that pairwise intersect in at most one point, we say that the linear re-embedding unfolds the edge .
The notion of -discriminants for configurations of points in was introduced and further developed by Gelfand, Kapranov and Zelevinsky in . We present the theory in its original formulation for Laurent polynomials. In our applications we only deal with polynomials with non-negative exponents defined over .
Throughout this section, we let be an algebraically closed field. We fix a configuration of points in , and a Laurent polynomial supported on :
We use the multiplicative notation if .
For generic choices of coefficients , the polynomial has no singularities in the algebraic torus . However, for special choices of coefficients, singularities do appear. Such special situations (and their algebraic closure) are determined by the ideal , where
It can be shown that whenever is a principal ideal, its unique generator is irreducible and can be defined over . The -discriminant is the unique (up to sign) irreducible polynomial with integer coefficients in the unknowns defining . If is not principal, we set and refer to as a defective configuration.
Assume . Then, the discriminant of the trapezoid in Figure 2 equals the Sylvester resultant of the univariate polynomials and . In particular, when we obtain
The same formulas hold if we pick any configuration of lattice points in containing all four vertices of the trapezoid, after replacing the corresponding variables among by zero.
Since -discriminants are invariant under affine transformations of the lattice , we may assume that the trapezoid has vertices and . Furthermore, is not a pyramid, so we know the planar configuration is not defective. We fix a polynomial with support on the given trapezoid, and compute its two partial derivatives:
Let be a general point where the discriminant vanishes. Then, admits a singular point in the torus . In particular, and so . Thus, both and have a common solution , so . We conclude that divides . Since both polynomials are irreducible over , the result follows.
Now, let and write . Assume is a singular point of in the torus . From (2.12) we conclude that and we can use the equation to find the value of . Since, in addition, we obtain
as we wanted to show.
Conversely, if the right-hand side of (2.11) vanishes, then any point constructed from the vanishing of the partials (2.12) is a singularity of . The singularity lies in the torus if and only if . The latter is an open condition in the coefficients , and it is independent on the variable . Since the bottom expression in (2.13) has degree 1 in , we can find a unique