THE MODULI SPACE OF n POINTS ON THE LINE IS CUT OUT BY SIMPLE QUADRICS WHEN n IS NOT SIX

BENJAMIN HOWARD, JOHN MILLSON, ANDREW SNOWDEN AND RAVI VAKIL

Abstract. A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This note deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any linearization, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold (the space of six points with equal weight). Unlike many facts in geometric invariant theory, these results (at least for the stable locus) are field-independent, and indeed work over the integers.

Contents

1. Introduction 1 2. The invariants of n points on ℙ^{1} as a graphical algebra 4 3. An analysis of a neighborhood of a strictly semistable point 14 4. Proof of Main Theorem 19 References 25

1. Introduction

We consider the space of n (ordered) points on the projective line, up to automorphisms of the line. In characteristic 0, the best description of this is as the Geometric Invariant Theory quotient (ℙ^{1})^{n}//SL(2). This is one of the most classical examples of a GIT quotient, and is one of the first examples given in any course (see [MS, §2], [MFK, §3], [N, §4.5], [D, Ch. 11], [DO, Ch. I], ...). We find it remarkable that well over a century after this fundamental invariant theory problem first arose, we have so little understanding of its defining equations, when they should be expected to be quite simple.

The construction of the quotient depends on a linearization, in the form of a choice of weights of the points w = (w_{1}, …, w_{n}) (the weight vector). We denote the resulting

Date: Friday, July 14, 2006. The first two authors were partially supported by NSF grant DMS-0104006 and the last author was

partially supported by NSF Grant DMS-0228011.

2000 Mathematics Subject Classification: Primary 14L24, Secondary 14D22, 14H10.

quotient M_{w}. A particularly interesting case is when all points are treated equally, when w = (1, …, 1) = 1^{n}. We call this the equilateral case because of its interpretation as the moduli space of equilateral space polygons in symplectic geometry (e.g. [HMSV, §2.3]). At risk of confusion, we denote this important case M_{1^{n}} by M_{n} for simplicity. We say that n points of ℙ^{1} are w-stable (resp. w-semistable) if the sum of the weights of points that coincide is less than (resp. no more than) half the total weight. The dependence on w will be clear from the context, so the prefix w- will be omitted. The n points are strictly semistable if they are semistable but not stable. Then M_{w} is a projective variety, and GIT gives a natural projective embedding. The stable locus of M_{w} is a fine moduli space for the stable points of (ℙ^{1})^{n}. The strictly semistable locus of M_{w} is a finite set of points, which are the only singular points of M_{w}. The question we wish to address is: what are the equations of M_{w}?

We prefer to work as generally as possible, over the integers, so we now define the moduli problem of stable n-tuples of points in ℙ^{1}. For any scheme B, families of stable n-tuples of points in ℙ^{1} over B are defined to be n morphisms to ℙ^{1} such that no more than half the weight is concentrated at a single point of ℙ^{1}. More precisely, a family is a morphism (ϕ_{1}, …, ϕ_{n}) : B × {1, …, n} → ℙ^{1} such that for any I ⊂ {1, …, n} such that ∑_{i∈I} w_{i} ≥ ∑_{i=1}^{n} w_{i}/2, we have ∩_{i∈I}ϕ_{i}^{−1}(p) = ∅ for all p. Then there is a fine moduli space for this moduli problem, quasiprojective over ℤ, which indeed has a natural ample line bundle, the one suggested by GIT. (This is well-known, but in any case will fall out of our analysis.) In parallel with GIT, we define stable points. Hence this space has extrinsic projective geometry. The question we will address in this context is: what are its equations?

The main moral of this note is taken from Chevalley’s construction of Chevalley groups: when understanding a vector space defined geometrically, choosing a basis may obscure its structure. Instead, it is better to work with an equivariant generating set, and equivariant linear relations. A prototypical example is the standard representation of 𝔖_{n}, which is best understood as the permutation representation on the vector space generated by e_{1}, …, e_{n} subject to the relation e_{1} + ⋯ + e_{n} = 0. As an example, we give yet another short proof of Kempe’s theorem (Theorem 2.3), which has been called the “deepest result” of classical invariant theory [Ho, p. 156] (in the sense that it is the only result that Howe could not prove from standard constructions in representation theory). Another application is the computation in §2.11 of the degree of all M_{w}.

We now state our main theorem. We will describe a natural (equivariant) set of “graphical” generators of the algebra of invariants (in Section 2). The algebraic structure of the invariants is particularly transparent in this language, and as an example we give a short proof of Kempe’s Theorem 2.3, and give an easy basis of the group of invariants (by “non-crossing variables”, Theorem 2.5). We then describe some geometrically (or combinatorially) obvious relations, the (linear) sign relations, the (linear) Plücker relations, and the (quadratic) simple binomial relations.

1.1. Main Theorem. — With the single exception of w = (1, 1, 1, 1, 1, 1), the following holds.

(a) Over a field of characteristic 0, the GIT quotient (ℙ^{1})^{n}//SL_{2} (with its natural projective embedding) is cut out scheme-theoretically by the sign, Plücker, and simple binomial relations.

(b) Over ℤ (or any base scheme), the fine moduli space of stable n-tuples of points on ℙ^{1} is quasiprojective over ℤ. Under its natural embedding, its closure is cut out by the sign, Plücker, and simple binomial relations.

The exceptional case w = (1, 1, 1, 1, 1, 1) is the Segre cubic threefold.

Note that Geometric Invariant Theory does not apply to SL(2)-quotients in positive characteristic, as SL(2) is not a reductive group in that case. Thus for (b) we must construct the moduli space by other means.

The idea of the proof is as follows. We first reduce the question to the equilateral case, where n is even. We do this by showing a stronger result, which reduces such questions about the ideal of relations of invariants to the equilateral case.

1.2. Theorem (reduction to equilateral case, informal statement). — For any weight w, there is a natural map from the graded ring of projective invariants for 1^{|w|} to those for w. Under this map, each of our generators for 1^{|w|} is sent to either a generator for w, or to zero. Moreover, the relations for 1^{|w|} generate the relations for w.

In Section 2.12, we will state this result precisely, and prove it, once we have introduced some terminology. This is a stronger result than we will need, as it refers to the full ring of projective invariants, not just to the quotient variety. It is also a stronger statement than simply saying that M_{w} is naturally a linear section of M_{|w|}.

We then verify Theorem 1.1 “by hand” in the cases n = 2m ≤ 8 (§2.14). The cases n = 6 and n = 8 are the base cases for our later argument (which is ironic, as in the n = 6 case the result does not hold!). In §4, we show that the result holds set-theoretically, and that the projective variety is a fine moduli space away from the strictly semistable points. The strictly semistable points are more delicate, as the quotient is not naturally a fine moduli space there; we instead give an explicit description of a neighborhood of a strictly semistable point, as the affine variety corresponding to rank one (n/2 − 1) × (n/2 − 1) matrices with entries distinct from 1, using the Gel’fand-MacPherson correspondence. We prove the result in this neighborhood in §3.

1.3. Related questions. Our question is related to another central problem in invariant theory: the invariants of binary forms, or equivalently n unordered points on ℙ^{1}, or equivalently equations for M_{n}/𝔖_{n}. These generators and relations are more difficult, and the relations are certainly not just quadratic. Mumford describes Shioda’s solution for n = 8 [Sh] as “an extraordinary tour de force” [MFK, p. 77]. One might dream that the case n = 10 might be tractable by computer, given the explicit relations for M_{10} described here.

We remark on the relation of this paper to our previous paper [HMSV]. That paper dealt with the ideal of relations among the invariants, and a central result was that this ideal was cut out by equations of degree at most 4. The present article deals with the moduli space as a projective variety; it is not at all clear that our quadrics generate the ideal of invariants.

Question. Do the simple binomial relations generate the ideal of relations among invariants (if n ≠ 6)?

By Theorem 1.2, it suffices to consider the equilateral case. [HMSV] suggests one approach: one could hope to show that the explicit generators given there lie in the ideal given by these simple binomial quadrics.

Even special cases are striking, and are simple to state but computationally too complex to verify even by computer. For example, we will describe particularly attractive relations for M_{10} of degree degrees 3, 5, 7 (§2.10); do these lie in the ideal of our simple quadrics? We will describe a quadric relation for M_{12} (§2.6). Is this a linear combination of our simple binomial quadrics?

These questions lead to more speculation. The shape of the proof of the main theorem suggests an explanation for the existence of an exception: when the number of points gets “large enough”, the relations are all inherited from “smaller” moduli spaces, so at some point (in our case, when n = 8) the relations “stabilize”.

Speculative question. We know that M_{n} satisfies Green’s property N_{0} (projectively normality — Kempe’s Theorem 2.3), and the questions above suggest that M_{n} might satisfy property N_{1} (projectively normal and cut out by quadrics) for n > 6. Might it be true that each p, M_{n} satisfies N_{p} for p ≫ 0? (Property N_{2} means that the scheme satisfies N_{1}, and the syzygies among the quadrics are linear. These properties measure the “niceness” of the ideal of relations.)

Speculative question. Might similar results hold true for other moduli spaces of a similar flavor, e.g. \overline{M}_{0,n} or even \overline{M}_{g,n}? Are equations for moduli spaces inherited from equations of smaller moduli spaces for n ≫ 0? For further motivation, see the striking work of [KT] on equations cutting out \overline{M}_{0,n}. Also, an earlier example of quadratic equations inherited from smaller moduli spaces appeared in [BP], in the case of Cox rings of del Pezzo surfaces. I thank B. Hassett for pointing out this reference.

Acknowledgments. The last author thanks Lucia Caporaso, Igor Dolgachev, Diane Maclagan, Brendan Hassett, and Harm Derksen for useful suggestions.

2. The invariants of n points on ℙ^{1} as a graphical algebra

We give a convenient alternate description of the generators (as a group) of the ring of invariants of n ordered points on ℙ^{1}. By graph we will mean a directed graph on n vertices labeled 1 through n. Graphs may have multiple edges, but may have no loops. The multidegree of a graph Γ is the n-tuple of valences of the graph, denoted deg Γ. The bold font is a reminder that this is a vector. We consider each graph as a set of edges. For each edge e of Γ, let h(e) be the head vertex of e and t(e) be the tail. We use multiplicative

×  =

Figure 1. Multiplying (directed) graphs

notation for the “union” of two graphs: if Γ and Δ are two graphs on the same set of vertices, the union is denoted by Γ · Δ (so for example deg Γ + deg Δ = deg Γ · Δ), see Figure 1. (We will occasionally use additive and subtractive notation when we wish to “subtract” graphs. We apologize for this awkwardness.)

For each graph Γ, define X_{Γ} ∈ H^{0}((ℙ^{1})^{n}, O_{(ℙ^{1})^{n}}(deg Γ)) by

(1)  X_{Γ} = ∏_{edge e of Γ}(p_{h(e)} − p_{t(e)}) = ∏_{edge e of Γ}(u_{h(e)}v_{t(e)} − u_{t(e)}v_{h(e)}).

If S is a non-empty set of graphs of the same degree, [X_{Γ}]_{Γ∈S} denotes a point in projective space ℙ^{|S|−1} (assuming some such X_{Γ} is nonzero of course). For any such S, the map (ℙ^{1})^{n} −−→ [X_{Γ}]_{Γ∈S} is easily seen to be invariant under SL(2): replacing p_{i} by p_{i}+a preserves X_{Γ}; replacing p_{i} by ap_{i} changes each X_{Γ} by the same factor; and replacing p_{i} by 1/p_{i} also changes each X_{Γ} by the same factor.

The First Fundamental Theorem of Invariant Theory [D, Thm. 2.1] states that, given a weight w, the ring of invariants of (ℙ^{1})^{n}//SL_{2} is generated (as a group) by the X_{Γ} where deg Γ is a multiple of w. The translation to the tableaux is as follows. Choose any ordering of the edges e_{1}, …, e_{|Γ|} of Γ. Then X_{Γ} corresponds to any 2 × |Γ| tableau where the top row of the ith column is h(e_{i}) and the bottom row is t(e_{i}). We will soon see advantages of this graphical description as compared to the tableaux description.

We now describe several types of relations among the X_{Γ}, which will all be straightforward: the sign relations, the Plücker (or straightening) relations, the simple binomial relations, and the Segre cubic relation.

2.1. The sign (linear) relations. The sign relation X_{Γ·x⃗y} = −X_{Γ·y⃗x} (Figure 2) is immediate, given the definition (1). Because of the sign relation, we may omit arrowheads in identities where it is clear how to consistently insert them (see for example Figures 7 and 9, where even the vertices are implicit).

2.2. The Plücker (linear) relations. The identity of Figure 3 may be verified by direct calculation. If Γ is any graph on n vertices, and Δ_{1}, Δ_{2}, Δ_{3} are three graphs on the same vertices given by identifying the four vertices of Figure 3 with the some four of the n vertices of Γ, then

(2)  X_{Γ·Δ_{1}} + X_{Γ·Δ_{2}} + X_{Γ·Δ_{3}} = 0.

= −

Figure 2. An example of the sign relation

4 1  1 3  3 4  4 2 2 3  1  2 +  +  = 0

Figure 3. The Plücker relation for n = 4 (and w = (1, 1, 1, 1))

+  +  = 0

Figure 4. An example of a Plücker relation

These relations are called Plücker relations (or straightening rules). See Figure 4 for an example. We will sometimes refer to this relation as the Plücker relation for Γ · Δ_{1} with respect to the vertices of Δ_{1}.

Using the Plücker relations, one can reduce the number of generators to a smaller set, which we will do shortly (Proposition 2.4). However, a central thesis of this article is that this is the wrong thing to do too soon; not only does it obscure the 𝔖_{n} symmetry of this generating set, it also makes certain facts opaque. As an example, we give a new proof of Kempe’s theorem. The proof will also serve as preparation for the proof of the main theorem, Theorem 1.1.

2.3. Kempe’s Theorem [HMSV, Thm. 4.6]. — The lowest degree invariants generate the ring of invariants.

Note that the lowest-degree invariants are of weight ϵ_{w}w, where ϵ_{w} = 1 if |w| is even, and 2 if |w| is odd.

Γ  Γ′

Figure 5. Constructing Γ′ from Γ (example with w = (1, 1, 2, 2), d = 2)

Proof. We begin in the case when w = (1, …, 1) where n is even. Recall Hall’s Marriage Theorem: given a finite set of men M and women W, and some men and women are compatible (a subset of M × W), and it is desired to pair the women and men compatibly, then it is necessary and sufficient that for each subset S of women, the number of men compatible with at least one of them is at least |S|.

Given a graph Γ of multidegree (d, …, d), we show that we can find an expression Γ = ∑ ±Δ_{i} · Ξ_{i} where deg Δ_{i} = (1, …, 1). Divide the vertices into two equal-sized sets, one called the “positive” vertices and one called the “negative” vertices. This creates three types of edges: positive edges (both vertices positive), negative edges (both vertices negative), and neutral edges (one vertex of each sort). When one applies the Plücker relation to a positive edge and a negative edge, all resulting edges are neutral (see Figure 3, and take two of the vertices to be of each type). Also, each regular graph must have the same number of positive and negative edges. Working inductively on the number of positive edges, we can use the Plücker relations so that all resulting graphs have only neutral edges. We thus have an expression Γ = ∑ ±Γ_{i} where each Γ_{i} has only neutral edges and is hence a bipartite graph. Each vertex of Γ_{i} has the same valence d, so any set of p positive vertices must connect to at least p negative edges. By Hall’s Marriage Theorem, we can find a matching Δ_{i} that is a subgraph of Γ_{i}, with “residual graph” Ξ_{i} (i.e. Γ_{i} = Δ_{i} · Ξ_{i}). Thus the result holds in the equilateral case.

We next treat the general case. If |w| is odd, it suffices to consider the case 2w, so by replacing w by 2w if necessary, we may assume ϵ_{w} = 1. The key idea is that M_{w} is a linear section of M_{|w|}. Suppose deg Γ = dw. Construct an auxiliary graph Γ′ on |w| vertices, and a map of graphs π : Γ′ → Γ such that (i) the preimage of vertex i of Γ consists of w_{i} vertices of Γ′, (ii) π gives a bijection of edges, and (iii) each vertex of Γ′ has valence d, i.e. Γ′ is d-regular. (See Figure 5 for an illustrative example. There may be choice in defining Γ′). Then apply the algorithm of the previous paragraph to Γ′. By taking the image under π, we have our desired result for Γ. □

Choosing a planar representation makes termination of certain algorithms straightforward as well, as illustrated by the following argument. Consider the vertices of the graph to be the vertices of a regular n-gon, numbered (clockwise) 1 through n. A graph is said to be non-crossing if no two edges cross. Two edges sharing one or two vertices are considered not to cross. A variable X_{Γ} is said to be non-crossing if Γ is.

2.4. Proposition (graphical version of “straightening algorithm”). — For each w, the noncrossing variables of degree w generate ⟨X_{Γ}⟩_{deg Γ=w} (as an abelian group).

b d a f e c  y x z w

Figure 6. The triangle inequality implies termination of straightening: b + c > e, a + d > f

This is essentially the straightening algorithm (e.g. [D, §2.4]) in this situation.

Proof. We explain how to express X_{Γ} in terms of non-crossing variables. If Γ has a crossing, choose one crossing wx · yz (say Γ = wx · yz · Γ′), and use the Plücker relation (2)involving wxyz to express Γ in terms of two other graphs wy · xz · Γ′ and wz · xy · Γ′. Repeat this if possible. We now show that this process terminates, i.e. that this algorithm will express X_{Γ} in terms of non-crossing variables. Both of these graphs have lower sum of edgelengths than Γ (see Figure 6, using the triangle inequality on the two triangles with side lengths a, d, f and b, c, e). As there are finite number of graphs of weight w, and hence a finite number of possible sums of edge-lengths, the process must terminate. □

2.5. Theorem (non-crossing basis of invariants). — For each w, the non-crossing variables of degree w form a basis for ⟨X_{Γ}⟩_{deg Γ=w}.

Proof. Proposition 2.4 shows that the non-crossing variables span, so it remains to show that they are linearly independent. Assume otherwise that they are not always linearly independent, and thus there s a simplest nontrivial relation R for some smallest w (where the w are partially ordered by |w| and #w). The relation R states that some linear combination of these graphs is 0. Suppose this relation involves #w = n vertices. Then not all of the graphs in R contain edge (n − 1)n (or else we could remove one copy of the edge from each term in R and get a smaller relation).

Then identify vertices (n − 1) and n, throwing out the graphs containing edge (n − 1)n. This “contraction” gives a natural bijection between non-crossing graphs on n vertices with multidegree w, not containing edge (n − 1)n, and non-crossing graphs on n − 1 vertices with multidegree w′ (where w′_{n−1} = w_{n−1} + w_{n}). The resulting relation still holds true (in terms of invariants, we insert the relation p_{n−1} = p_{n} into the older relation). We have contradicted the minimality of R, so no such counterexample R exists. □

2.6. Binomial (quadratic) relations. We next describe some obvious binomial relations. If deg Γ_{1} = deg Γ_{2} and deg Δ_{1} = deg Δ_{2}, then clearly X_{Γ_{1}·Δ_{1}}X_{Γ_{2}·Δ_{2}} = X_{Γ_{1}·Δ_{2}}X_{Γ_{2}·Δ_{1}}. We call these the binomial relations. A special case are the simple binomial relations when deg Δ_{i} = (1, 1, 1, 1, 0, …, 0) = 1^{4}0^{n−4}, or some permutation thereof. Examples are shown in Figures 7 and 9.

=

Figure 7. A simple binomial relation for n = 5

Γ_{1} =  Γ_{2} =  Δ_{1} =  Δ_{2} =

Figure 8. The building blocks of Figure 7

(We have now defined all the relations relevant to the Main Theorem 1.1, so the reader is encouraged to reread its statement.)

In the even democratic case, the smallest quadratic binomial relations that are not simple binomial relations appear for n = 12, and

deg Γ_{i} = deg Δ_{i} = (1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0).

In the introduction, we asked if these quadratics are linear combinations of the simple binomial relations.

By the Plücker relations, the binomial relations are generated by those where the Γ_{i} and Δ_{i} are non-crossing, and similarly for the simple binomial relations. This restriction can be useful to reduce the number of equations, but as always, symmetry-breaking obscures other algebraic structures. (Note that even though we may restrict to the case where Γ_{i} and Δ_{j} are non-crossing, we may not restrict to the case where Γ_{i} · Δ_{j} are non-crossing, as the following examples with n = 5 and n = 8 show.)

As an example, consider the case n = 5 (with the smallest democratic linearization (2, 2, 2, 2, 2)). One of the simple binomial relations is shown in Figure 7. The building blocks Γ_{i} and Δ_{j} are shown Figure 8. In fact, these quadric relations cut out M_{5} in ℙ^{5}, as can be checked directly, or as follows from Theorem 1.1. The 𝔖_{5}-representation on the quadrics is visible.

2.7. As a second example, consider n = 8 (and the democratic linearization (1, …, 1)). Because there are \binom{8}{4}/2 = 35 ways of partitioning the 8 vertices into two subsets of size 4, and each such partition gives one simple binomial relation (where the Γ_{i} and Δ_{j} are non-crossing, see comments two paragraphs previous), we have 35 quadric relations on M_{8}, shown in Figure 9.

The space of quadric relations forms an irreducible 14-dimensional 𝔖_{8}-representation, which we show by representation theory. If V is the vector space of quadratic relations,

=

Figure 9. One of the simple binomial relations for n = 8 points

we have the exact sequence

0 → V → Sym^{2} H^{0}(M_{8}, O(1)) → H^{0}(M_{8}, O(2)) → 0

of representations. By counting non-crossing graphs, we can calculate h^{0}(M_{8}, O(2)) = 91 (the 8th Riordan number or Motzkin sum) and h^{0}(M_{8}, O(1)) = C_{4} = 14 (the 4th Catalan number), from which dim V = 14. As the representation H^{0}(M_{8}, O(1)) is identified, we can calculate the representation Sym^{2} H^{0}(M_{8}, O(1)), and observe that the only 14dimensional subrepresentations it contains are irreducible. (Simpler still is to compute the character of the 196-dimensional representation H^{0}(M_{8}, O(1))^{⊗2}, and decompose it into irreducible representations, using Maple for example, finding that it decomposes into representations of dimension 1 + 14 + 14 + 20 + 35 + 56 + 56; Sym^{2} H^{0}(M_{8}, O(1)) is of course a subrepresentation of this.)

As our quadric relations are nontrivial, and form an 𝔖_{8}-representation, we have given generators of the quadric relations. Necessarily they span the same vector space of the 14 relations given in [HMSV, §8.3.4]. Our relations have the advantage that the 𝔖_{8}-action is clear, but the major disadvantage that it is not a priori clear that the vector space they span has dimension 14. We suspect there is an 𝔖_{8}-equivariant description of the linear relations among the generators, but we have been unable to find one.

In [HMSV, §8.3.4] it was shown that the ideal of relations of M_{8} was generated by these fourteen quadrics, and hence by our 35 simple binomial quadrics. We will use this as the base case of our induction later.

2.8. The Segre cubic relation. Other relations are also clear from this graphical perspective. For example, Figure 10 shows an obvious relation for M_{6}, which is well-known to be a cubic hypersurface (the Segre cubic hypersurface, see for example [DO, p. 17] or [D, Example 11.6]). As this is a nontrivial cubic relation (this can be verified by writing it in terms of a non-crossing basis), it must be the Segre cubic relation. Interestingly, although the relation is not 𝔖_{6}-invariant, it becomes so modulo the Plücker relations (2). Note that there are no (nontrivial) binomial relations for M_{6} (which is cut out by this cubic), so the Segre relation cannot be in the ideal generated by the binomial relations.

2.9. Remark: Segre cubic relations for n ≥ 8. There are analogous cubic relations for n ≥ 8, by simply adding other vertices. The n = 8 case is given in Figure 11. For n ≥ 8, these Segre cubic relations lie in the ideal generated by the simple binomial relations. We will use this in the proof of Theorem 1.1. This follows from the case n = 8, which can be verified in a couple of ways. As stated above, [HMSV] shows that the ideal cutting out

=

Figure 10. The Segre cubic relation (graphical version)

=

Figure 11. The Segre cubic relation for n = 8

M_{8} is generated by the fourteen quadrics of [HMSV, §8.3.4], which by §2.7 is the ideal generated by the simple binomial relations, and the cubic lies in this ideal. One may also verify that the Segre relation lies in the ideal generated by the fourteen quadrics by explicit calculation (omitted here).

2.10. Other relations. There are other relations, that we will not discuss further. For example, consider the democratic case for n even. Then 𝔖_{n} acts on the set of graphs. Choose any graph Γ. Then

∑_{σ∈𝔖_{n}} sgn(σ)X_{σ(Γ)}^{i} = 0

is a relation for i odd and 1 < i < n − 1. Reason: substituting for X’s in terms of p’s (or more correctly the u’s and v’s) using (1)to obtain an expression E, and observing that 𝔖_{n} acts oddly on E, we see that we must obtain a multiple of the Vandermonde, which has degree (n − 1, …, n − 1) > deg E. Hence E = 0. It is not clear that this is a nontrivial relation, but it appears to be so in small cases. In particular, the case n = 6, i = 3 is the Segre cubic relation. In the introduction, we asked if these relations for n = 10 lie in the ideal generated by the simple binomial quadric relations.

2.11. Degree of the GIT quotient. As an application of these coordinates, we compute the degree of all M_{w}. For example, we will use this to verify that the degree is 1 when |w| = 6 and w ≠ (1, …, 1), although this can also be done directly.

We would like to intersect the moduli space M_{w} with n − 3 coordinate hyperplanes of the form X_{Γ} = 0 and count the number of points, but these hyperplanes will essentially never intersect properly. Instead, we note that each hyperplane X_{Γ} = 0 is reducible, and consists of a finite number of components of the form M_{w′} where the number of points #w′ is n − 1. We can compute the multiplicity with which each of these components appears. The algorithm is then complete, given the base case n = 4. Here, more precisely, is the algorithm.

(a) (trivial case) If n = 3, the moduli space is a point, so the degree is 1.

(b) (base case) If w = (d, d, d, d), then deg M_{w} = d, as the moduli space is isomorphic to ℙ^{1}, embedded by the d-uple Veronese. (This may be seen by direct calculation, or by noting that a base-point-free subset of those variables of degree (d, d, d, d) are “dth powers” of variables of degree (1, 1, 1, 1).)

(c) If n > 4 and w satisfies w_{j}+w_{k} ≤ ∑ w_{i}/2 for all j, k, we choose any Γ of weight w. We can understand the components of X_{Γ} = 0 by considering the morphism π : (ℙ^{1})^{n} −U_{w} → [X_{Γ}]_{deg Γ=w}, where U_{w} is the unstable locus. Directly from the formula for X_{Γ}, we see that for each pair of vertices j, k with an edge joining them, such that w_{j} + w_{k} < ∑ w_{i}/2, there is a component that can be interpreted as M_{w′}, where w′ is the same as w except that w_{j} and w_{k} are removed, and w_{j} + w_{k} is added (call this w_{0} for convenience). We interpret this as removing vertices j and k, and replacing them with vertex 0. This component corresponds to the divisor

(3)  (u_{j}v_{k} − u_{k}v_{j})^{m_{jk}} = 0

on the source of π, where m_{jk} is the number of edges joining vertices j and k. If Δ is the reduced version of this divisor, u_{j}v_{k}−u_{k}v_{j} = 0, then the correspondence between between Δ → M_{w} and (ℙ^{1})^{n−1}−U_{w′} → M_{w′} is as follows. For each Γ′ of degree w′, we lift X_{Γ′} to any X_{Γ} where Γ is a graph on {1, …, n} of degree w whose “image” in {1, …, n} ∪ {0} \ {j, k} is Γ′. (In other words, to w_{j} of the w_{0} edges meeting vertex 0 in Γ′, we associate edges meeting vertex j in Γ, and similarly with j replaced by k.) If Γ′′ is any other lift, then X_{Γ} = ±X_{Γ′′} on Δ, because using the Plücker relations, X_{Γ} ± X_{Γ′′} can be expressed as a combination of variables containing edge jk, which all vanish on Δ.

From (3), the multiplicity with which this component appears is m_{ij}, the number of edges joining vertices j and k.

If w_{j} + w_{k} = ∑ w_{i}/2, then M_{w′} is a strictly semistable point, and of dimension 0 smaller than dim M_{w} − 1, and hence is not a component. (Our base case is n = 4, not 3, for this reason.)

(d) If n ≥ 4 and there are j and k such that w_{j} + w_{k} > ∑ w_{i}/2, then the rational map (ℙ^{1})^{4} −−→ M_{w} has a base locus. Any graph X_{Γ} of degree w necessarily contains a copy of edge jk, so (u_{j}v_{k} − u_{k}v_{j}) is a factor of any of the X_{Γ}. Hence M_{w} is naturally isomorphic to M_{w−e_{j}−e_{k}}, so we replace w by w − e_{j} − e_{k}, and repeat the process. Note that if n = 4, then the final resulting quadruple must be of the form (d, d, d, d).

For example, deg M_{4} = 1, deg M_{6} = 3, deg M_{8} = 40, and deg M_{10} = 1225 were computed by hand. (This appears to be sequence A012250 on Sloane’s On-line encyclopedia of integer sequences [Sl].) The calculations deg M_{6} = 3 and deg M_{2,2,2,2,2} = 5 are shown in Figure 12. and 13 respectively. At each stage, w is shown, as well as the Γ used to calculate the next stage. In these examples, there is essentially only one such w′ at each stage, but in general there will be many. The vertical arrows correspond to identifying components of X_{Γ} (step (c)). The first arrow in Figure 12 is labeled ×3 to point out the reader that the next stage can be obtained in three ways. The degrees are obtained inductively from the bottom up.

2 Γ =  deg = 3 w = (1, 1, 1, 1, 1, 1) (c) ×3 (2, 1, 1, 1, 1) (2, 2, 1, 1) (c)  (1, 1, 1, 1) deg = 1  deg = 1 (d)  (b)

Figure 12. Computing deg M_{6} = 3 (recall that M_{6} is the Segre cubic threefold S_{3})

Γ =  deg = 5 (c) (4, 2, 2, 2) w = (2, 2, 2, 2, 2) ×5  (3, 2, 2, 1)  (2, 2, 1, 1)  (1, 1, 1, 1) deg = 1 (d)  (d)  (d)  (b)

Figure 13. Computing deg M_{(2,2,2,2,2)} = 5 using an inconvenient choice of Γ (recall that M_{(2,2,2,2,2)} is a degree 5 del Pezzo surface)

(The reader is encouraged to show that deg M_{8} = 40, and that this algorithm indeed gives deg M_{dw} = d^{n−3} deg M_{w}.)

2.12. Reduction of the Main Theorem 1.1 to the equilateral case. We next show that the Main Theorem 1.1 in the equilateral case (when w = (1, …, 1)) implies the Main Theorem 1.1 in general. The argument is similar in spirit to our proof of Kempe’s Theorem 2.3.

Consider the commutative diagram:

\xymatrix{0 \ar[r] & I_{n} \ar[r] \ar[d]^{α} & ⊕_{Γ}ℤX_{Γ} \ar[r]^{ϕ} \ar[d]^{β} & R_{n} \ar[d]^{γ} \ar[r] & 0 \\ 0 \ar[r] & I_{w} \ar[r] & ⊕_{Ω}ℤX_{Ω} \ar[r]^{ψ} & R_{w} \ar[r] & 0,}

where R_{n} (resp. R_{w}) is the ring of invariants for 1^{n} (resp. w), the Γ’s range over matchings of n vertices, n = |w| is even, and the Ω’s range over multi-valence w graphs. The map β takes X_{Γ} to X_{Ω} where Ω is given by identifying vertices of Γ within the same clump; if a loop is introduced then it maps to zero. (We take the clumps to be subsets of adjacent vertices in the n-gon.)

2.13. Theorem. — The map α : I_{n} → I_{w} is surjective.

Proof. It is clear that β is surjective (and hence γ too). Thus by the five lemma, it suffices to prove that ker β surjects onto ker γ.

We have that R_{n} = ⊕_{G}ℤ · ϕ(X_{G}), where G ranges over regular non-crossing graphs on {1, …, n}. (In other words, such ϕ(X_{G}) form a ℤ-basis of R_{n}.) Similarly R_{w} = ⊕_{H}ℤ·ψ(X_{H}) as H ranges over non-crossing graphs of multi-valence a multiple of w. For each H, there is exactly one non-crossing G for which γ(ϕ(X_{G})) = ψ(X_{H}). Thus ker(γ) = ⊕_{G}ℤ · ϕ(X_{G}) where the sum is over those non-crossing G which contain least one edge which connects two vertices in a single clump. Fix such a G, with an edge a → b where a and b are in the same clump. We will show that X_{G} ∈ ker β.

Partition {1, …, n} into two equal sized subsets A and B (“positive” and “negative”) such that a ∈ A, and b ∈ B. As in the proof of Kempe’s theorem 2.3, we can write X_{G} = ∑_{i} ±X_{G_{i}}, where the G_{i} are (possibly crossing) graphs each containing the edge a → b. (The process described in the proof of Theorem 2.3 involves trading a pair of edges, one positive and one negative, for two neutral edges. No neutral edges such as a → b are affected by this process.)

By applying Hall’s marriage theorem (repeatedly) to each G_{i}, we can write X_{G_{i}} = ∏_{j=1}^{k} ϕ(X_{Γ_{i,j}}), where the Γ_{i,j} are matchings, and k = deg(X_{G}). (At least) one Γ_{i,j} contains the edge a → b, so, β(∏_{j=1}^{k} X_{Γ_{i,j}}) = 0. Hence β(X_{G}) = β(∑ ±X_{G_{i}}) = 0 as desired. □

2.14. Verification of the Main Theorem 1.1 in small cases. The cases |w| = 2 and |w| = 4 are trivial.

If |w| = 6 and w ≠ (1, …, 1), then w = (3, 2, 1), (2, 2, 2), (2, 2, 1, 1), or (2, 1, 1, 1, 1). The first two cases are points, and the next two cases were verified to have degree 1 in §2.11 (see Figure 12).

The case w = (1, 1, 1, 1, 1, 1, 1, 1) was verified in §2.7, so by §2.12, the case |w| = 8 follows.

Thus the cases |w| ≥ 10 remain.

3. An analysis of a neighborhood of a strictly semistable point

We now show the result in a neighborhood of a strictly semistable point, in the equilateral case w = 1^{n=2m}, in characteristic 0, by explicitly describing an affine neighborhood of such a point. This affine neighborhood has a simple description: it is the space of (m − 1) × (m − 1) matrices of rank at most 1, where no entry is 1 (Lemma 3.3). The strictly semistable point corresponds to the zero matrix.

3.1. The Gel’fand-MacPherson correspondence: the moduli space as a quotient of the Grassmannian. We begin by recalling the Gel’fand-MacPherson correspondence, an alternate description of the moduli space. The Plücker embedding of the Grassmannian G(2, n) ↪ ℙ^{\binom{n}{2}−1} is via the line bundle O(1) that is the positive generator for Pic G(2, n).

This generator may be described explicitly as follows. Over G(2, n), we have a tautological exact sequence of vector bundles

(4)  \xymatrix{0 \ar[r] & S \ar[r] & O^{⊕n} \ar[r] & Q \ar[r] & 0}

where S is the tautological rank 2 subbundle (over [Λ] ∈ G(2, n), it corresponds to Λ ⊂ ℂ^{n}), and Q is the tautological rank n − 2 quotient bundle. Then ∧^{2}S = O(1) is a line bundle, and is the dual to O(1). Dualizing (4)we get a map ∧^{2}O^{⊕n} → ∧^{2}S^{∗}. Then ∧^{2}S^{∗} can be easily checked to be be generated by these global sections. We call these sections s_{ij}, and note that they satisfy the following relations: the sign relations s_{ij} = −s_{ji} (so s_{ii} = 0) inherited from ∧^{2}O^{⊕n}, and the Plücker relations

s_{ij}s_{kl} − s_{ik}s_{jl} − s_{jk}s_{il} = 0.

These equations cut out the Grassmannian in ℙ^{\binom{n}{2}−1}.

The connection to n points on ℙ^{1} is as follows. Given a general point of the Grassmannian corresponding to the subspace Λ of n-space, we obtain n points on ℙ^{1}, by considering the intersection of Λ with the n coordinate hyperplanes, and projectivizing. This breaks down if Λ is contained in a coordinate hyperplane. (The point [Λ] is GIT-stable if the resulting n points in ℙ^{1} are GIT-stable, and similarly for semistable. We recover the crossratio of four points via s_{ij}s_{kl}/s_{il}s_{jk}.)

Let D(s_{1n}) be the distinguished open set where s_{1n} ≠ 0. (In the correspondence with marked points, this corresponds to the locus where the first point is distinct from the last point.) Then D(s_{1n}) is isomorphic to 𝔸^{2(n−2)}, with good coordinates as follows. Given Λ ∉ D(s_{1n}), choose a basis for Λ, written as a 2 × n matrix. As Λ ∉ D(s_{1n}), the first and last columns are linearly independent, so up to left-multiplication by GL(2), there

is a unique way to choose a basis where the first column is [ \begin{matrix}0 \\ 1 \end{matrix} ] and the last column

is [ \begin{matrix}1 \\ 0 \end{matrix} ]. We choose the “anti-identity” matrix rather than the identity matrix, because

we will think of the first column as [0; 1] ∈ ℙ^{1}, and the last column as [1; 0]. (Another interpretation is as follows. If Λ is interpreted as a line in ℙ^{n−1}, and H_{1}, …, H_{n} are the coordinate hyperplanes, then if Λ does not meet H_{1} ∩ H_{n}, then it meets H_{1} at one point of H_{1} − H_{1} ∩ H_{n} ≅ ℂ^{n−2} and H_{n} at one point of H_{n} − H_{1} ∩ H_{n} ≅ ℂ^{n−2}, and Λ is determined by these two points. The coordinates on the first space are the x’s, and the coordinates on the second are the y’s.)

Thus if the 2 × n matrix is written

[ \begin{matrix}0 & x_{2} & x_{3} & ⋯ & x_{n−1} & 1 \\ 1 & y_{2} & y_{3} & ⋯ & y_{n−1} & 0 \end{matrix} ]

then we have coordinates x_{2}, …, x_{n−1}, y_{2}, …, y_{n−1} on our affine chart. For convenience, we define x_{1} = 0, y_{1} = 1, x_{n} = 1, y_{n} = 0.

Under the trivialization (O(1), s_{1n})|_{D(s_{1n})} ≅ (O, 1)|_{D(s_{1n})}, in these coordinates, the section s_{ij} may be interpreted as

s_{ij} = x_{j}y_{i} − x_{i}y_{j}.

We can use this to immediately verify the Plücker relations. We also recover the x_{i} and y_{j} from the sections via

(5)  x_{i} = s_{1i}/s_{1n}  y_{j} = s_{jn}/s_{1n}.

The Grassmannian has dimension 2(n − 2) = 2n − 4. To obtain our moduli space, we take the quotient of G(2, n) by the maximal torus T ⊂ SL(2, n), which has dimension n − 1. (Thus as expected the quotient has dimension n − 3.) We will write elements of this maximal torus as λ = (λ_{1}, ..., λ_{n}). To describe the linearization, we must describe how λ acts on each s_{ij}: λ_{i} acts on s_{ij} with weight 1, and on the rest of the s_{ij}’s by weight 0. This action certainly preserves our relations.

Then we can see how to construct the quotient as a Proj: the terms that have weight (d, d, …, d) correspond precisely to d-regular graphs on our n vertices. Hence we conclude that this projective scheme is precisely the GIT quotient of n points on the projective line, as the graded rings are the same. This is the Gel’fand-MacPherson correspondence. The relations we have described on our X_{Γ} clearly come from the relations on the Grassmannian. (That is of course no guarantee that we have them all!)

3.2. A neighborhood of a strictly semistable point. Let π : G(2, 2m)^{ss} → M_{w} be the quotient map. Let p be a strictly semistable point of the moduli space M_{w}, without loss of generality the image of (0, …, 0, ∞, …, ∞). We say an edge ij on vertices {1, …, 2m} is good is i ≤ m < j (if it “doesn’t connect two 0’s or two ∞’s”). We say a graph on {1, …, 2m} is good all of its edges are good. We say an edge or graph is bad if it is not good. Let P be the set of good matchings of {1, …, 2m}. Let

U_{P} = {q ∈ M_{w} : X_{Γ}(q) ≠ 0 for all Γ ∈ P }.

(In the dictionary to n points on ℙ^{1}, this corresponds to the set where none of the first m points is allowed to be the same as any of the last m points.) Note that p ∈ P, and π^{−1}(U_{P}) ⊂ D(s_{1,2m}).

3.3. Lemma. — U_{P} is an affine variety, with coordinate ring generated by W_{ij} and Z_{ij} (1 < i ≤ m < j < 2m) with relations

(6)  W_{ij}W_{kl} = W_{il}W_{kj}

(i.e. the matrix [W_{ij}] has rank 1) and

(7)  Z_{ij}(W_{ij} − 1) = 1

(i.e. the matrix [W_{ij}] has no entry 1).

This has a simple interpretation: U_{P} is isomorphic to the space of (m − 1) × (m − 1) matrices of rank at most 1, where each entry differs from 1, and p is the unique singular point, corresponding to the zero matrix. (We remark that this is the cone over the Segre embedding of ℙ^{m−1} × ℙ^{m−1}.) Hence we have described a neighborhood of the singular point rather explicitly.

Proof. Let V_{P} = {[Λ] ∈ G(2, 2m) : s_{ij}(Λ) ≠ 0 for all i ≤ m < j }, so V_{P} = π^{−1}(U_{P}). Then V_{P} is an open subset of D(s_{1,2m}). In terms of the coordinates on D(s_{1,2m}) ≅ ℂ^{4m−4} described above, V_{P} is described by

(8)  x_{j}y_{i} − x_{i}y_{j} ≠ 0,  x_{j} ≠ 0,  y_{i} ≠ 0

for i ≤ m < j. Let T be the maximal torus T ⊂ SL(2m). By our preceding discussion, using (5), λ_{k} (1 < k < 2m) acts on x_{k} and y_{k} with weight 1, and on the other x_{l} and y_{l}’s with weight 0. The torus λ_{1} acts on x_{l} with weight 0, and on y_{l} with weight −1. The torus λ_{2m} acts on x_{l} with weight −1, and on y_{l} with weight 0.

We analyze the quotient V_{P} /T by writing T as a product, T = T′T′′, and restricting to invariants of T′, then of T′′. Let T′ be the subtorus of T such that λ_{1} = λ_{2m}, and T′′ be the subtorus (λ, 1, …, 1, λ^{−1}). We take the quotient first by T′. It is clear that the invariants are given by u_{i} = x_{i}/y_{i} for 1 < i ≤ m and v_{j} = y_{j}/x_{j} for m + 1 ≤ j < 2m. From (8), the quotient of V_{P} is cut out by the inequality u_{i}v_{j} − 1 ≠ 0.

T′′ acts on this quotient as follows: λ acts on u_{i} by weight −2, and v_{j} by weight 2. The invariants of this quotient by T′′ are therefore generated by u_{i}v_{j} and 1/(u_{i}v_{j} − 1).

Hence if we take W_{ij} = u_{i}v_{j}, the invariants are generated by the W_{ij}, subject to the relations that the matrix [W_{ij}] has rank 1, and also W_{ij} ≠ 1. □

Now let I ⊂ ℂ[{X_{Γ}}] be the ideal of relations of the invariants of M_{w}, and let I_{V} be the ideal generated by the linear Plücker relations and the simple binomial relations. We have already shown that I_{V} ⊂ I.

Let S be the multiplicative system of monomials in X_{Γ} generated by those X_{Γ} where Γ ∈ P.

3.4. Theorem. — If n = 2m ≥ 8, then S^{−1}I_{V} = S^{−1}I. In other words, the sign, Plücker, and simple binomial relations cut out the moduli space on this open subset.

As the Main Theorem 1.1 is true for n = 2m = 8 (§2.7), this theorem holds in that “base” case.

Proof. By Γ we will mean a general matching, and by Δ, we will mean a matching in P. We have a surjective map

ℂ[{X_{Γ}/X_{Δ}}]/S^{−1}I_{V} → ℂ[{X_{Γ}/X_{Δ}}]/S^{−1}I

(that we wish to show is an isomorphism), and Lemma 3.3 provides an isomorphism

ℂ[{X_{Γ}/X_{Δ}}]/S^{−1}I ≅ O(U_{P}) ≅ ℂ[{W_{ij}, Z_{ij}}]/J_{W Z},

where J_{W Z} ⊂ ℂ[{W_{ij}}, {Z_{ij}}] is the ideal generated by the relations (6)and (7).

By comparing the moduli maps, we see that this isomorphism is given by

(9)  W_{ij} ↦ \frac{X_{1i·j(2m)·Γ}}{X_{1j·i(2m)·Γ}} ,  Z_{ij} ↦ \frac{X_{1j·i(2m)·Γ}}{X_{1(2m)·ij·Γ}}

where Γ is any matching on {1, …, 2m} − {1, i, j, 2m} such that 1j · i(2m) · Γ ∈ P. (By the simple binomial relations, this is independent of Γ.) The description of the isomorphism in the reverse direction is not so pleasant, and we will spend much of the proof avoiding describing it explicitly.

We thus have a surjective morphism

ψ : ℂ[{X_{Γ}/X_{Δ}}] → ℂ[{W_{ij}, Z_{ij}}]/J_{W Z}

whose kernel is S^{−1}I, which contains S^{−1}I_{V}. We wish to show that the kernel is S^{−1}I_{V}. We do this as follows. For each 1 < i ≤ m < j < 2m, fix a matching Γ_{i,j} on {1, …, 2m} − {1, i, j, 2m} so that 1j · i(2m) · Γ_{i,j} ∈ P. Consider the subring of ℂ[{X_{Γ}/X_{Δ}}] generated by

w_{ij} = \frac{X_{1i·j(2m)·Γ_{i,j}}}{X_{1j·i(2m)·Γ_{i,j}}} , z_{ij} = \frac{X_{1j·i(2m)·Γ_{i,j}}}{X_{1(2m)·ij·Γ_{i,j}}} .

(Compare this to (9).) Call this subring ℂ[{w_{ij}, z_{ij}}]/J_{wz}.

The proof consists of two steps. Step 1. We show that any element of ℂ[{X_{Γ}/X_{Δ}}] differs from an element of ℂ[{w_{ij}, z_{ij}}]/J_{wz} by an element of S^{−1}I_{V}. We do this in several smaller steps. Step 1a. We show that any X_{Γ}/X_{Δ} can be written as a linear combination of X_{Δ′}/X_{Δ} (where Δ′ is also good). Step 1b. We show that any such X_{Δ′}/X_{Δ} may be expressed (modulo S^{−1}I_{V}) in terms of X_{ik·jl·Γ}/X_{il·jk·Γ}, where i, j ≤ m < k, l, and Γ is good. Step 1c. We show that any such expression can be written (modulo S^{−1}I_{V}) in terms of w_{ij} and z_{ij}, i.e. modulo S^{−1}I_{V}, any such expression lies in ℂ[{w_{ij}, z_{ij}}]/J_{wz}.

Step 2. The kernel of the map ψ : ℂ[{w_{ij}, z_{ij}}]/J_{wz} → ℂ[{W_{ij}, Z_{ij}}]/J_{W Z} (given by w_{ij} ↦ W_{ij}, z_{ij} ↦ Z_{ij}) lies in S^{−1}I_{V}.

We now execute this strategy.

Step 1a. We first claim that X_{Γ}/X_{Δ} (Δ ∈ P) is a linear combination of units X_{Δ′}/X_{Δ} (i.e. Δ′ ∈ P) modulo the Plücker relations (the linear relations, which are in S^{−1}I_{V}). We prove the result by induction on the number of bad edges. The base case (if all edges of Γ are good, i.e. Γ ∈ P) is immediate. Otherwise, Γ has at least two bad edges, say ij and kl, where i, j ≤ m < k, l. Then X_{Γ} = ±X_{Γ−{ij,kl}+{ik,jl}} ± X_{Γ−{ij,kl}+{il,jk}} is a Plücker relation, and the latter two terms have two fewer bad edges, completing the induction.

Step 1b. We show that any element X_{Δ′}/X_{Δ} of ℂ[{X_{Γ}/X_{Δ}}] (Δ′ good) is congruent (modulo S^{−1}I_{V}) to an element of the form X_{ik·jl·Γ}/X_{il·jk·Γ}, where i, j ≤ m < k, l, and Γ is good. We prove this by induction on m. If m = 4, the result is true (§2.7). Assume now that m > 4. If Δ′ and Δ share an edge e, then let \overline{Δ′} and \overline{Δ} be the graphs on 2m−2 vertices obtained by removing this edge e. Then by the inductive hypothesis, the result holds for X\overline{Δ′}/X\overline{Δ}. By taking the resulting expression, and “adding edge e to the subscript of each term”, we get an expression for X_{Δ′}/X_{Δ}. Finally, if Δ′ and Δ share no edge, suppose in Δ′, 1 is connected to (m + 1); in Δ, 1 is connected to (m + 2); and in Δ′, (m + 2) is connected to 2. This is true after suitable reordering. Say Δ′ = 1(m + 1) · 2(m + 2) · Γ′ and Δ = 1(m + 2) · Γ. Then

\frac{X_{Δ′}}{X_{Δ}} = \frac{X_{1(m+2)·2(m+1)·Γ′}}{X_{1(m+2)·Γ}} · \frac{X_{1(m+1)·2(m+2)·Γ′}}{X_{1(m+2)·2(m+1)·Γ′}} .

For each factor of the right side, the numerator and the denominator “share an edge”, so we are done.

Step 1c. We next show that any such can be written (modulo S^{−1}I_{V}) in terms of w_{ij} and z_{ij}, i.e. modulo S^{−1}I_{V} lies in ℂ[{w_{ij}, z_{ij}}]/J_{wz}. If 2m = 8, the result again holds (§2.7). Assume now that 2m > 8. Given any X_{ik·jl·Γ}/X_{ij·kl·Γ} as in Step 1b, we will express it (modulo S^{−1}I_{V}) in terms of w_{ij} and z_{ij}. By the simple binomial relation (i.e. modulo S^{−1}I_{V}), we may assume that Γ is any good matching on {1, …, 2m} − {i, j, k, l}, and in particular that there are edges ab, cd ∈ Γ such that {1, 2m} ⊂ {a, b, c, d, i, j, k, l}. Then the result for m = 4 implies that we can write X_{ik·jl·ab·cd}/X_{ij·kl·ab·cd} can be written in terms of w_{ij} and z_{ij} (in terms of the “m = 4 variables”). By taking this expression, and “adding in the remaining edges of Γ”, we get the desired result for our case.

Step 2. We will show that the kernel of the map ψ : ℂ[{w_{ij}, z_{ij}}]/J_{wz} → ℂ[{W_{ij}, Z_{ij}}]/J_{W Z} (given by w_{ij} ↦ W_{ij}, z_{ij} ↦ Z_{ij}) lies in S^{−1}I_{V}.

In order to do this, we need only verify that the relations (6)and (7)are consequences of the relations in S^{−1}I_{V}.

We first verify (6). By the simple binomial relation, we may write

(10)  W_{ij} = \frac{X_{1i·j(2m)·kl·Γ}}{X_{1j·i(2m)·kl·Γ}} ,  W_{kl} = \frac{X_{1k·l(2m)·ij·Γ}}{X_{1l·k(2m)·ij·Γ}} ,

(11)  W_{il} = \frac{X_{1i·l(2m)·jk·Γ}}{X_{1l·i(2m)·jk·Γ}} ,  W_{jk} = \frac{X_{1j·k(2m)·il·Γ}}{X_{1k·j(2m)·il·Γ}} .

We thus wish to show that modulo S^{−1}I_{V}, the product of the terms in (10) equals the product of the terms in (11). Choose any edge e ∈ Γ. The analogous question with m = 4, with Γ − e “removed from the subscripts”, is true (§2.7). Hence, by “adding Γ − e back in to the subscripts”, we get the analogous result here.

We next verify (7):

1 − W_{i,j} = \frac{X_{1j·i(2m)·Γ_{i,j}}}{X_{1j·i(2m)·Γ_{i,j}}} − \frac{X_{1i·j(2m)·Γ_{i,j}}}{X_{1j·i(2m)·Γ_{i,j}}} ≡ \frac{X_{1(2m)·ij·Γ_{i,j}}}{X_{1j·i(2m)·Γ_{i,j}}} = 1/Z_{i,j} (mod S^{−1}I_{V})

where the equivalence uses a linear Plücker relation. □

4. Proof of Main Theorem

We have reduced to the equilateral case w = 1^{n}, n = 2m, where n ≥ 10.

The reader will notice that we will use the simple binomial relations very little. In fact we just use the inductive structure of the moduli space: given a matching Δ on n − k of n vertices (4 ≤ k < n), and a point [X_{Γ}]_{Γ} of V_{n}, then either these X_{Γ} with Δ ⊂ Γ are all zero, or [X_{Γ}]_{Δ⊂Γ} satisfies the Plücker and simple binomial relations for k, and hence is a point of V_{k} if k ≠ 6. (The reader should think of this rational map [X_{Γ}] −−→ [X_{Γ}]_{Δ⊂Γ} as a forgetful map, remembering only the moduli of the k points.) In fact, even if k = 6 (and n ≥ 8), the point must lie in M_{6}, as the simple binomial relations for n > 6 induce the Segre cubic relation (§2.9). The central idea of our proof is, ironically, to use the case n = 6, where the Main Theorem 1.1 doesn’t apply.

We will call such Δ, where the X_{Γ} with Δ ⊂ Γ are not all zero and the corresponding point of M_{6} is stable, a stable (n − 6)-matching. One motivation for this definition is that given a stable configuration of n points on ℙ^{1}, there always exists a stable (n−6)-matching. (Hint: Construct Δ inductively as follows. We say two of the n points are in the same clump if they have the same image on ℙ^{1}. Choose any y in the largest clump, and any z in the second-largest clump; yz is our first edge of Δ. Then repeat this with the remaining vertices, stopping when there are six vertices left.) Caution: This is false with 6 replaced by 4!

Main Theorem 1.1 is a consequence of the following two statements, and Theorem 3.4. Indeed, (I) and (II) show Theorem 1.1 set-theoretically, and scheme-theoretically away from the strictly semistable points (in characteristic 0), and Theorem 3.4 deals with (a neighborhood of) the strictly semistable points.

(I) There is a natural bijection between points of V_{n} with no stable (n − 6)-matching, and strictly semistable points of M_{n}.

(II) If B is any scheme, there is a bijection between morphisms B → V_{N} missing the “no stable (n−6)-matching” locus (i.e. missing the strictly semistable points of M_{n}, by (I)) and stable families of n points B × {1, …, n} → ℙ^{1}. (In other words, we are exhibiting an isomorphism of functors.)

One direction of the bijection of (I) is immediate. The next result shows the other direction.

4.1. Claim. — If [X_{Γ}]_{Γ} is a point of V_{n} (n ≥ 10) having no stable (n − 6)-matching, then [X_{Γ}]_{Γ} is a strictly semistable stable point of M_{n}.

Several of the steps will be used in the proof of (II). We give them names so they can be referred to later.

Proof. Our goal is to produce a partition of n into two subsets of size n/2, such that the point of M_{n} given by this partition is our point of V_{n}. Throughout this proof, partitions will be assumed to mean into two equal-sized subsets.

We work by induction. We will use the fact that the result is also true for n = 6 (tautologically) and n = 8 (as V_{8} = M_{8}, §2.7).

Fix a matching Δ such that X_{Δ} ≠ 0. By the inductive hypothesis, each edge xy yields a strictly semistable point of M_{n−2}, and hence a partition of {1, …, n} − {x, y}, by considering all matchings containing xy. Thus for each xy ∈ Δ, we get a partition of {1, …, n} − {x, y}. If wx, yz are two edges of Δ, then we get the same induced partition of {1, …, n}− {w, x, y, z} (from the inductive hypothesis for n − 4), so all of these partitions arise from a single partition {1, …, n} = S_{0} ∐ S_{1}.

4.2. Δ two-overlap argument. As this partition is determined using any two edges of Δ, we would get the same partition if we began with any Δ′ sharing two edges with Δ, such that X_{Δ′} ≠ 0.

Defining the map to ℙ^{1}. Define ϕ : S_{0} ∐ S_{1} = {1, …, n} → ℙ^{1} by S_{0} → 0 and S_{1} → 1. For each matching Γ, define X′_{Γ} using these points of ℙ^{1}. Rescale (or normalize) all the X_{Γ} so X′_{Δ} = X_{Δ}. We will show that X′_{Γ} = X_{Γ} for all Γ, which will prove Claim 4.1.

4.3. One-overlap argument. For any Γ sharing an edge xy with Δ, X′_{Γ} = X_{Γ}, for the following reason: [X_{Ξ}]_{xy∈Ξ} lies in M_{n−2} by the inductive hypothesis, and this point of M_{n−2} corresponds to the map ϕ (as the partition S_{0} ∐ S_{1} was determined using this point of M_{n−2}), so [X_{Ξ}]_{xy∈Ξ} = [X′_{Ξ}]_{xy∈Ξ}, and the normalization X′_{Δ} = X_{Δ} ≠ 0 ensures that X′_{Ξ} = X_{Ξ} for all Ξ containing xy.

4.4. Reduction to Γ with X′_{Γ} ≠ 0. It suffices to prove the result for those graphs Γ, all of whose edges connect S_{0} and S_{1} (i.e. no edge is contained in S_{0} or S_{1}; equivalently, X′_{Γ} ≠ 0). We show this by showing that any X_{Γ} is a linear combination of such graphs, by induction on the number i of edges of Γ contained in S_{0} (= the number contained in S_{1}). The base case i = 0 is tautological. For the inductive step, choose an edge wx ∈ Γ contained in S_{0} and an edge yz contained in S_{1}. Then the Plücker relation using Γ and wxyz (with appropriate signs depending on the directions of edges) is

±X_{Γ} ± X_{Γ−wx−yz+wy+xz} ± X_{Γ−wx−yz+wz+xy} = 0,

and both Γ − wx − yz + wy + xz and Γ − wx − yz + wz + xy have i − 1 edges contained in S_{0}, and the result follows.

4.5. pqrs argument, first version. Finally, assume that X′_{Γ} ≠ 0 and that Γ shares no edge with Δ. See Figure 14. Let qr be an edge of Γ (so ϕ(q) ≠ ϕ(r)), and let pq and rs be edges of Δ containing q and r respectively (so ϕ(p) ≠ ϕ(q) and ϕ(r) ≠ ϕ(s)). Then ϕ(p) ≠ ϕ(s), as ϕ takes on only two values. Let Δ′ = Δ − pq − rs + qr + ps, so X′_{Δ′} ≠ 0 as ϕ(q) ≠ ϕ(r) and ϕ(p) ≠ ϕ(s). Then X_{Δ′} = X′_{Δ′} by the one-overlap argument 4.3, as Δ′ shares an edge with Δ (indeed all but two edges), so X_{Δ′} ≠ 0. Hence by the Δ two-overlap argument 4.2, Δ′ defines the same partition S_{0} ∐ S_{1}, and hence the same map ϕ : {1, …, n} → ℙ^{1}. Finally, Γ shares an edge with Δ′, so X′_{Γ} = X_{Γ} by the one-overlap argument 4.3.

We have thus completed the proof of Claim 4.1. □

Proof of (II). The result boils down to the following desideratum: Given any (n − 6)- matching Δ on some {1, …, n} − {a, b, c, d, e, f}, there should be a bijection between:

(a) morphisms π : B → V_{n} contained in the open subset where Δ is a stable (n − 6)- matching, and

(b) stable families of points ϕ : B × {1, …, n} → ℙ^{1} where ϕ|_{B×{a,...,f}} is also a stable family, and for any edge xy of Δ, ϕ|_{B×{x}} does not intersect ϕ|_{B×{y}}.

p q  s Δ  Γ t r Δ Γ

Figure 14. The pqrs argument (vertex t is used in §4.10)

We have already described the map (b) ⇒ (a) in §2. We now describe the map (a) ⇒ (b), and verify that (a) ⇒ (b) ⇒ (a) is the identity. (It will then be clear that (b) ⇒ (a) ⇒ (b) is the identity: given a stable family of points parameterized by B, we get a map from B to an open subset of M_{n}, which is a fine moduli space, hence (b) ⇒ (a) is an injection. The result then follows from the fact that (a) ⇒ (b) ⇒ (a) is the identity.)

We work by induction on n. The case n = 8 was checked earlier (§2.7).

The map to ℙ^{1}. Given an element of (a), define a family of n points of ℙ^{1} (an element of (b)) as follows. (i) ϕ : B × {a, …, f} → ℙ^{1} is given by the corresponding map B → M_{6}. (ii) If yz is an edge of Δ, we define B × ({1, …, n} − {y, z}) → ℙ^{1} extending (i) by considering the matchings containing yz, which by the inductive hypothesis give a point of M_{n−2}. (iii) The morphisms of (ii) agree “on the overlap”, as given two edges wx and yz of Δ, we get B ×({1, …, n}−{w, x, y, z}) → ℙ^{1} by considering the matchings containing wx·yz, which by the inductive hypothesis give a map to M_{n−4}. Here we are using that n ≥ 10; and if n = 10, we need the fact that the Segre cubic relation cutting out M_{6} is induced by the quadrics cutting out M_{n} for n ≥ 8 (Remark 2.9). Thus we get a well-defined morphism ϕ : B × {1, …, n} → ℙ^{1}.

4.6. Δ two-overlap argument, cf. §4.2. If Δ′ is another matching on {1, …, n} − {a, …, f} sharing at least 2 edges with Δ, with X_{Δ′·Ξ} ≠ 0 for some matching Ξ of {a, …, f}, we obtain the same ϕ, as ϕ can be recovered by considering only two edges of Δ when using (ii).

Defining X′. Define X′_{Γ} for all matchings Γ using ϕ and the moduli morphism of eqn. (1). The coordinates X_{Γ} are projective (i.e. the set of X_{Γ} is defined only up to scalars); scale them so that X_{Δ·Ξ} = X′_{Δ·Ξ} for all matchings Ξ of {a, …, f}. Note that if xy is an edge of Δ, then ϕ(x) ≠ ϕ(y), as there exists a matching Ξ of {a, …, f} such that X′_{Δ·Ξ} ≠ 0.

The following result will confirm that (a) ⇒ (b) ⇒ (a) is the identity, concluding the proof of (II).

4.7. Claim. — We have the equality X_{Γ} = X′_{Γ} for all Γ.

Proof. This proof will occupy us until the end of §4.14.

4.8. One-overlap argument. As in §4.3, the result holds for those Γ sharing an edge yz with Δ: by considering only those variables X_{Γ′} containing the edge yz (including both X_{Γ} and X_{Δ}), we obtain a point of M_{n−2}. This point of M_{n−2} is the one given by ϕ (this was part of how ϕ was defined), so [X_{Γ′}]_{yz∈Γ′} = [X′_{Γ′}]_{yz∈Γ′}. By choosing a matching Ξ on {a, …, f} so that X_{Δ·Ξ} ≠ 0, we have that X_{Γ}X′_{Δ·Ξ} = X_{Δ·Ξ}X′_{Γ}. Using X_{Δ·Ξ} = X′_{Δ·Ξ} ≠ 0, we have X_{Γ} = X′_{Γ}, as desired.

We now deal with the remaining case, where Γ and Δ share no edge.

4.9. Reduction to Γ with X′_{Γ} ≠ 0 (cf. §4.4). It suffices to prove the result for those graphs such that X′_{Γ} ≠ 0, or equivalently that for each edge xy of Γ, ϕ(x) ≠ ϕ(y). We show this by showing that any X_{Γ} is a linear combination of such graphs, by induction on the number of edges xy of Γ with ϕ(x) = ϕ(y). For the purposes of this paragraph, call these bad edges. The base case i = 0 is tautological. For the inductive step, choose a bad edge wx ∈ Γ (with ϕ(w) = ϕ(x)), and another edge yz such that ϕ(y), ϕ(z) ≠ ϕ(w). (Such an edge exists, as by stability, less than n/2 elements of {1, …, n} take the same value in ℙ^{1}.) Then the Plücker relation using Γ with respect to wxyz is

±X_{Γ} ± X_{Γ−wx−yz+wy+xz} ± X_{Γ−wx−yz+wz+xy} = 0,

and both Γ − wx − yz + wy + xz and Γ − wx − yz + wz + xy have at most i − 1 bad edges, and the result follows.

Recall that we are proceeding by induction. We first deal with the case n ≥ 14, assuming the cases n = 10 and n = 12. We will then deal with these two stray cases. This logically backward, but the n ≥ 14 case is cleaner, and the two other cases are similar but more ad hoc.

4.10. The case n ≥ 14. pqrs argument, second version. As n ≥ 14, there is an edge qr of Γ not meeting abcdef. See Figure 14. By §4.9, we may assume ϕ(q) ≠ ϕ(r). Let pq and rs be the edges of Δ meeting q and r respectively (so ϕ(p) ≠ ϕ(q) and ϕ(r) ≠ ϕ(s). (i) (cf. the similar argument of §4.5) If ϕ(p) ≠ ϕ(s), then let Δ′ = Δ − pq − rs + qr + ps; then Δ′ defines the same family of n points as Δ by the two-overlap argument §4.6, and Γ and Δ′ share an edge, so we are done by the one-overlap argument §4.8. (More precisely, this argument applies on the open subset of B where ϕ(p) ≠ ϕ(s).) (ii) If ϕ(p) = ϕ(s), then ϕ(p) ≠ ϕ(r). (More precisely, this argument applies on the open set where ϕ(p) ≠ ϕ(r).) Let st be the edge of Γ containing s. (It is possible that t = p.) Let Γ′ = Γ − qr − st + rs + qt and Γ′′ = Γ − qr − st + qs + rt be the other two terms in the Plücker relation for Γ for pqrs. Then Γ′ shares edge rs with Δ, so X′_{Γ′} = X_{Γ′} by the one-overlap argument §4.8, and by applying (i) to Γ′′ (swapping the names of r and s), X′_{Γ′′} = X_{Γ′′}, so by the Plücker relation, X′_{Γ} = X_{Γ} as desired.

4.11. The cases n = 10 and n = 12. We are assuming that Γ and Δ share no edges. If there is an edge of Γ not meeting {a, …, f}, the pqrs-argument §4.10 applies, so assume otherwise. Divide {1, …, n} into two subsets abcdef and ghij (resp. ghijkl) if n = 10 (resp. n = 12), where the edges of Δ are gh, ij, and (if n = 12) kl. By renaming abcdef, we

b  d  f j  e c a  Γ Γ Γ Γ Δ  Δ Γ g  h  i

Figure 15. The problematic graphs for n = 10

g  b  d  f i  j  e c a  Γ  Γ Γ Γ Δ  Γ  Γ Δ k h  Δ  l

Figure 16. The problematic graphs for n = 12

may assume the edges of Γ are ag, bh, ci, dj, and either ef (if n = 10, see Figure 15) or ek and fl (if n = 12, see Figure 16).

4.12. Suppose that ϕ(a) ≠ ϕ(b). Note that we will only use that ag, bh ∈ Γ, gh ∈ Δ, and ϕ(a) ≠ ϕ(b) — we will use this argument again below. There is a matching Ξ of cdef so that if xy ∈ Ξ, then ϕ(x) ≠ ϕ(y). (This is a statement about stable configurations of 6 points on ℙ^{1}: if we have a stable set of 6 points on ℙ^{1}, then no three of them are the same point. Hence for any four of them cdef, we can find a matching of this sort.) Let Δ′ = Ξ · ab · Δ. Then by the simple binomial relations (our first invocation!) X_{Δ′}X_{Γ} = X_{Δ′−ab−gh+ag+bh}X_{Γ+ab+gh−ag−bh} and X′_{Δ′}X′_{Γ} = X′_{Δ′−ab−gh+ag+bh}X′_{Γ+ab+gh−ag−bh}. However, by the one-overlap argument §4.8, X_{Δ′} = X′_{Δ′} ≠ 0 (Δ′ and Δ share edge ij), X_{Δ′−ab−gh+ag+bh} = X′_{Δ′−ab−gh+ag+bh} (Δ′ − ab − gh + ag + bh and Δ share edge ij), and X_{Γ+ab+gh−ag−bh} = X′_{Γ+ab+gh−ag−bh} (Γ + ab + gh − ag − bh and Δ share edge gh), so we are done.

We are left with the case ϕ(a) = ϕ(b).

4.13. Suppose now that n = 10. As ϕ(a) = ϕ(b), ϕ(b) is distinct from ϕ(e) and ϕ(f) (as ϕ(a), …, ϕ(f) are a stable set of six points on ℙ^{1}). By the Plücker relations for Γ (using agef),

±X_{Γ} ± X_{Γ−ag−ef+ae+gf} ± X_{Γ−ag−ef+af+eg} = 0,

and similarly for the X′ variables. By applying the argument of §4.12 with e and a swapped, we have X′_{Γ−ag−ef+af+eg} = X_{Γ−ag−ef+af+eg}, and by applying the argument of §4.12 with f and a swapped, we have X′_{Γ−ag−ef+ae+gf} = X_{Γ−ag−ef+ae+gf}, from which X′_{Γ} = X_{Γ}, concluding the n = 10 case.

4.14. Suppose finally that n = 12. If ϕ(c) ≠ ϕ(d), we are done (by the same argument as §4.12, with ab replaced by cd), and similarly if ϕ(e) ≠ ϕ(f). Hence the only case left is if ϕ(a) = ϕ(b), ϕ(c) = ϕ(d), and ϕ(e) = ϕ(f), and (by stability of the 6 points ϕ(a), …, ϕ(f)) these are three distinct points of ℙ^{1}. Consider the Plücker relation for Γ with respect to bhci. One of the other two terms is Γ−bh−ci+bi+ch, and X′_{Γ−bh−ci+bi+ch} = X_{Γ−bh−ci+bi+ch} (by the same argument as in §4.12, as ϕ(a) ≠ ϕ(c)). We thus have to prove that X_{Γ′} = X′_{Γ′} for the third term in the Plücker relation, where

Γ′ = ag · bc · hi · dj · ek · fl.

For this, apply the argument of §4.13, with abghef replaced by felkbc respectively. □

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Benjamin Howard: Department of Mathematics, University of Maryland, College Park, MD 20742, USA, bhoward@math.umd.edu

John Millson: Department of Mathematics, University of Maryland, College Park, MD 20742, USA, jjm@math.umd.edu

Andrew Snowden: Department of Mathematics, Princeton University, Princeton, NJ 08544, USA, asnowden@math.princeton.edu

Ravi Vakil: Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA, vakil@math.stanford.edu