Exterior algebra methods for the Minimal Resolution Conjecture

by

David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer, Charles Walter

Abstract: If r ≥ 6, r ≠ 9, we show that the Minimal Resolution Conjecture fails for a general set of γ points in ℙ^{r} for almost \frac{1}{2} \sqrt{r} values of γ. This strengthens the result of Eisenbud and Popescu [1999], who found a unique such γ for each r in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud and Schreyer [2000] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the Minimal Resolution Conjecture fails for linearly normal curves of degree d and genus g when d ≥ 3g − 2, g ≥ 4, reproving results of Schreyer, Green, and Lazarsfeld.

1 Introduction

From the Hilbert function of a homogeneous ideal I in a polynomial ring S over a field k one can compute a lower bound for the graded betti numbers

β_{i,j} = dim_{k} Tor_{i}^{S}(S/I, k)_{j}

because these numbers are the graded ranks of the free modules in the minimal free resolution of I. The graded betti numbers are upper semicontinuous in families of ideals with constant Hilbert function, and in many cases this lower bound is achieved by the general member of such a family. The statement that it is achieved for a particular family T is called the Minimal Resolution Conjecture (MRC) for T, following Lorenzini, who made the conjecture for the family of ideals of sufficiently general sets of γ points in ℙ^{r} (all r, γ).

The minimal resolution conjecture for a general set of γ points in ℙ^{r} has received considerable attention; see Eisenbud and Popescu [1999] for full references and discussion. In particular, it is known that the minimal resolution conjecture is satisfied if r ≤ 4 (Gaeta [1951] and [1995], Geramita-Lorenzini [1989], Ballico-Geramita [1986], Walter [1995], Lauze [1996]), or γ ≫ r (Hirschowitz-Simpson [1996]), but computer evidence produced by Schreyer, extended also by Mats Boij in his 1994 thesis and by Beck and Kreuzer in [1996] suggested that it might fail for certain examples starting with 11 points in ℙ^{6}. Indeed, Eisenbud and Popescu [1999] proved that if r ≥ 6, r ≠ 9, then the minimal resolution conjecture fails for a general set of

γ = r + ⌊(3 + \sqrt{8r + 1})/2⌋

points in ℙ^{r} .

The first and the second authors are grateful to the NSF for support during the preparation of this work.

This family of counterexamples to the minimal resolution conjecture begins with Schreyer’s suggested 11 points in ℙ^{6}; and it seems to have contained all the examples in print in 1999. However, the thesis of Boij, completed in 1994 but only published in [2000], contains computations suggesting one further counterexample: 21 points in ℙ^{15}.

In this paper we will simplify and extend the idea of Eisenbud and Popescu to prove the existence of infinitely many new counterexamples, including the one suggested by Boij. We prove:

Theorem 1.1 The Minimal Resolution Conjecture fails for the general set of γ points in ℙ^{r} whenever r ≥ 6, r ≠ 9, and

r + 2 + \sqrt{r + 2} ≤ γ ≤ r + (3 + \sqrt{8r + 1})/2,

and also when (r, γ) = (8, 13) or (15, 21).

Thus we get about (\sqrt{2} − 1) \sqrt{r} counterexamples for each r ≥ 6, r ≠ 9.

Our proof not only gives more, but it avoids the subtle degeneration argument used by Eisenbud and Popescu. In the presentation below we will skip some of the steps presented in Eisenbud-Popescu [1999], and fully treat only the new ideas, so it may be helpful to the reader to explain their strategy and the point at which it differs from ours. Their proof can be divided into three steps, starting from a set Γ of γ general points in ℙ^{r}:

I. Their crucial first step is to consider the Gale transform of Γ ⊂ ℙ^{r}, which is a set Γ′ of γ points in ℙ^{s} with γ = r + s + 2 determined “naively” as follows: If the columns of the (r + 1) × γ matrix M represent the points of Γ, then a (s+1)×γ matrix N representing Γ′ is given by the transpose of the kernel of M (see Eisenbud-Popescu [1999] for a precise definition). The free resolution of the canonical modules of the homogeneous coordinate rings of Γ and Γ′ are related, and using this relation and the fact that (under our numerical hypotheses) the ideal of Γ′ is not 2-regular, they construct a map ϕ_{Γ} from a certain linear free complex F_{•}(μ) to the dual of the resolution of the ideal I_{Γ} of Γ. For the r and γ considered in the theorem, straightforward arithmetic shows that the injectivity of the top degree component of ϕ_{Γ} would force the graded betti numbers of I_{Γ} to be too large for the minimal resolution conjecture to hold. (We will recall the definition of F_{•}(μ) in Section 3 below.)

II. They show that ϕ_{Γ} is an injection of complexes if F_{•}(μ) has a property somewhat weaker than exactness called linear exactness or irredundancy (see Definition 2.1 below).

III. For a generic set of points Γ they show that F_{•}(μ) is irredundant by a subtle degeneration argument: they degenerate the general set Γ to lie on a special curve C, and the argument finishes with a study of a refined stability property of the tangent bundle of the projective space restricted to C in a certain embedding of C connected with the Gale transform of the points.

In our new proof step I remains unchanged, but II is replaced by a more precise statement, which requires us to verify a weaker condition in III; this weak condition can be verified without any degeneration argument, making the replacement for step III much simpler. More precisely the injectivity of ϕ_{Γ} is only needed for a sufficiently large irredundant quotient of the complex F_{•}(μ), and we give a criterion for this weaker condition using exterior algebra methods. The new criterion is checked in step III with an argument inspired by Mark Green’s proof [1999] of the Linear Syzygy Conjecture. The generality of Γ enters via work of Kreuzer [1994] showing that certain multiplication maps of the canonical module of the cone over Γ are 1-generic in the sense of Eisenbud [1988].

In Section 2 we introduce and study the irredundancy of linear complexes using the Bernstein-Gel’fand-Gel’fand (BGG) correspondence. In Section 3 we construct the complex whose irredundancy is the key to the failure of the minimal resolution conjecture. In the final Section 4 we briefly explain how Gale duality allows this theory to be applied to sets of points, and give the arithmetic part of the proof. As another application of our techniques we recover the result on curves of Schreyer (unpublished) and Green-Lazarsfeld [1988]; see Theorem 4.1.

It is a pleasure to thank Tony Iarrobino and the Mathematisches Forschungsinstitut Oberwolfach. Iarrobino, while reviewing Boij [2000] for MathSci, noticed that Boij’s example of 21 points in ℙ^{15} was not in the Eisenbud-Popescu list. The authors of the present paper were at that moment attending a conference at Oberwolfach. Iarrobino triggered our collaboration by asking Eisenbud and Popescu whether their methods could encompass the new example. The joy and success we had in answering his question owe much to the marvelous facilities and atmosphere in the Black Forest!

Notation: Throughout this paper k denotes an arbitrary field. Let V be a finitedimensional vector space over k. We work with graded modules over S := Sym(V) and E := ∧V^{∗}. We think of elements of V as having degree 1 and elements of V^{∗} as having degree −1. We write 𝔪 for the maximal ideal generated by V in S.

2 Linear complexes and exterior modules

In this section we illustrate the exterior algebra approach to linear free resolutions with some basic ideas used in the rest of the paper, and with a new proof of the linear rigidity theorem of Eisenbud and Popescu [1999].

Definition 2.1 A complex of graded free S-modules

F_{•} : ⋯ ^{ϕ_{3}} F_{2} ^{ϕ_{2}} F_{1} ^{ϕ_{1}} F_{0}

is a called a standard linear free complex if F_{i} is generated in degree i for all i (note that all the differentials ϕ_{i} are then represented by matrices of linear forms.) A linear free complex is any twisted (or shifted) standard linear free complex (that is, a right bounded complex of free modules F_{i} so that for some integer s the generators of F_{i} are all in degree s + i for all i). We say that a linear free complex F_{•} is irredundant (Eisenbud-Schreyer [2000]) or linearly exact (Eisenbud-Popescu [1999]) if the induced maps

F_{i+1}/𝔪F_{i+1} → 𝔪F_{i}/𝔪^{2}F_{i}

are monomorphisms for i > 0.

One sees immediately that a standard linear free complex F_{•} as above is irredundant if and only if H_{i}(F_{•})_{i} = 0 for all i > 0.

As explained in Eisenbud-Schreyer [2000], irredundancy can be conveniently rephrased via the Gel’fand-Gel’fand-Bernstein correspondence as follows:

If P is a graded E-module we define a linear free S-complex F_{•} = L(P) with free modules F_{i} = S ⊗_{k} P_{i} generated in degree i and differentials

ϕ_{i} : F_{i} → F_{i−1} 1 ⊗ p ↦ ∑ x_{i} ⊗ e_{i}p ∈ S ⊗ P_{i−1},

where {x_{i}} and {e_{i}} are (fixed) dual bases of V and V^{∗}. Up to twisting and shifting every linear free complex of S-modules arises from a unique E-module in this way. The linear complex L(P) is standard if P_{i} = 0 for all i < 0.

Given a graded E-module P we write P^{∗} for the dual graded E-module P^{∗} = Hom_{k}(P, k). We take k in degree 0, so that (P^{∗})_{i} is the dual vector space to P_{−i}.

Proposition 2.2 If

F_{•} = L(P) : ⋯  F_{2}  F_{1}  F_{0}

is a standard linear free complex of S-modules, then F_{•} is irredundant if and only if P^{∗} is generated as an E-module in degree 0.

Thus any linear free complex F_{•} = L(P) as in Proposition 2.2 has a unique maximal irredundant quotient F′_{•}, which is functorial in F_{•}, constructed as follows: If Q is the E-submodule of P^{∗} generated by P_{0}^{∗}, then F′_{•} := L(Q^{∗}). Alternatively, F′_{•} = L(P/N), where N is the submodule with graded pieces N_{i} = {n ∈ P_{i} | ∧^{i}V^{∗} · n = 0}. Observe that we have F′_{0} = F_{0}.

The next lemma is the basic tool which will allow us to deduce that a minimal free resolution is large. A weaker version of the result was implicit in Eisenbud-Popescu [1999].

Lemma 2.3 Suppose that α_{•} : F_{•} → G_{•} is a map from an irredundant standard linear free complex to a minimal free complex. If α_{0} is a split monomorphism, then α_{i} is a split monomorphism for all i ≥ 0.

Proof. By induction it suffices to prove that α_{1} is a split monomorphism. Since both F_{•} and G_{•} are minimal (that is, the differentials are represented by matrices of elements of 𝔪) there is a commutative diagram

[figure: F_{1}/𝔪F_{1}; 𝔪F_{0}/𝔪^{2}F_{0}; G_{1}/𝔪G_{1}; \overline{α}_{1}; 𝔪G_{0}/𝔪^{2}G_{0}]

whose vertical maps are induced by α_{•} and whose horizontal maps are induced by the differentials of F_{•} and G_{•}.

Since α_{0} makes F_{0} a summand of G_{0}, the right hand vertical map is a monomorphism. Because F_{•} is irredundant, the top map is a monomorphism. It follows that the map \overline{α}_{1} : F_{1}/𝔪F_{1} → G_{1}/𝔪G_{1} induced by α_{1} is a monomorphism, and since G_{1} is free, this implies that α_{1} is split.

Lemma 2.3 may be applied to give a lower bound on betti numbers:

Proposition 2.4 Suppose that α_{•} : F_{•} → G_{•} is a map from a standard linear complex to a minimal free resolution. There exists a map β_{•} : F_{•} → G_{•} which is homotopic to α_{•} and such that β_{•} factors through the maximal irredundant quotient F′_{•} of F_{•}. Further, if α_{0} : F_{0} → G_{0} is a split monomorphism, then the rank of G_{i} is at least as big as the rank of F′_{i}, for all i.

Proof. Let γ_{•} : F_{•} → F′_{•} be the natural map to the maximal irredundant quotient. The image of F′_{1} in F′_{0} = F_{0} is contained in that of F_{1}, so there is a map β_{1} : F′_{1} → G_{1} lifting α_{0} : F′_{0} → G_{0}. Since G_{•} is acyclic we may continue to lift, and so inductively we get a map of complexes β′_{•} : F′_{•} → G_{•}. We take β_{•} = β′_{•}γ_{•} and thus we have β_{0} = α_{0}. Since F_{•} is a free complex and G_{•} is acyclic, this implies that α_{•} is homotopic to β_{•}.

The second statement follows by applying Lemma 2.3 to the map β′_{•}.

In Eisenbud-Popescu [1999] a rigidity result for irredundancy is required. The proof given there is just a reference to the result on the rigidity of Tor proved by Auslander and Buchsbaum in [1958]. The exterior method yields a novel proof of this result (which we will not need in the sequel):

Proposition 2.5 Let R be a graded commutative or anti-commutative ring, and let M be a graded R-module which is generated by M_{0}. If R′ ⊂ R is a graded subring such that R′_{1} · M_{0} = M_{1}, then M is generated by M_{0} also as an R′-module.

Proof. We show by induction on n that R′_{1} · M_{n} = M_{n+1}, the initial case n = 0 being the hypothesis. If R′_{1} · M_{n−1} = M_{n}, then R′_{1} · M_{n} = R′_{1} · (R_{1}M_{n−1}) = R_{1} · (R′_{1}M_{n−1}) = R_{1} · M_{n} = M_{n+1}, where the second equality holds by the (anti)- commutativity assumption and the third by the induction hypothesis.

Corollary 2.6 (Linear Rigidity) Let S = k[x_{0}, …, x_{r}] be a polynomial ring, and let

F_{•} : ⋯  F_{2}  F_{1}  F_{0}

be an irredundant standard linear free complex. Let S′ = S/I be a graded quotient of S. The complex G_{•} := S′ ⊗_{S} F_{•} is irredundant iff H_{1}(G_{•}) is zero in degree 1.

Proof. The irredundancy of G_{•} involves only information about S′_{0} and S′_{1}, so we may assume that S′ is a polynomial ring S′ = Sym(V′) with V′ = V/I_{1}. Write E′ = ∧(V^{′∗}) ⊂ E = ∧(V^{∗}) for the corresponding exterior algebras. If F_{•} = L(P), then the complex G_{•} over S′ corresponds to the same graded vector space P, regarded as an E′-module by restriction of scalars. Consider now the truncated complex

K_{•} : ⋯  0  0  G_{1}  G_{0}.

The hypothesis H_{1}(G_{•})_{1} = 0 implies that H_{1}(K_{•})_{1} = 0 and hence, by Proposition 2.2, that P_{1}^{∗} = V^{′∗} · P_{0}^{∗}. On the other hand, P^{∗} is generated as an E-module by P_{0}^{∗} since F_{•} is an irredundant standard linear complex. So by Proposition 2.5 we have also P^{∗} = E′ · P_{0}^{∗}. By Proposition 2.2 this suffices.

Remark 2.7 A similar “linear rigidity” result for complexes over the exterior algebra can be also deduced from Proposition 2.5 applied this time for R a polynomial ring.

3 The complexes F_{•}(μ)

We will apply Proposition 2.2 and Lemma 2.3 to a family of linear complexes F_{•}(μ) (which were called E^{−1}(μ) in Eisenbud-Popescu [1999]). It is convenient here to define them in terms of the functor L.

Let U, V, and W be finite-dimensional vector spaces of dimensions u, v, and w, respectively. Let A := ∑_{l} ∧^{l}W^{∗} ⊗ Sym_{l}(U^{∗}), and let Q := ∑_{l} ∧^{l+1}W^{∗} ⊗ Sym_{l}(U^{∗}). Then A is an anticommutative graded algebra, and Q is a graded A-module. If μ : W ⊗ U → V is a pairing, then the dual μ^{∗} : V^{∗} → W^{∗} ⊗ U^{∗} extends to a map of algebras μ̃ : E = ∧(V^{∗}) → A. We regard Q as an E-module via μ̃, and we consider the corresponding linear free complex F_{•}(μ) := L(Q^{∗}) over S = Sym(V):

F_{•}(μ) : 0 → ∧^{w}W ⊗ D_{w−1}(U) ⊗ S(−w + 1) → ⋯ → ∧^{2}W ⊗ U ⊗ S(−1) → W ⊗ S,

where D_{m}(U) denotes the m-graded piece of the divided power algebra.

Though we shall not need the formula, it is easy to give the differentials explicitly: δ_{l}(μ) : F_{l}(μ) → F_{l−1}(μ), is the composite of the tensor product of the diagonal maps of the exterior and divided powers algebras

∧^{l+1}W ⊗ D_{l}U ⊗ S(−l) ∧^{l} W ⊗ W ⊗ D_{l−1}U ⊗ U ⊗ S(−l),

and the map induced by the pairing μ

∧^{l}W ⊗ W ⊗ D_{l−1}U ⊗ U ⊗ S(−l) ∧^{l} W ⊗ D_{l−1}U ⊗ S(−l + 1).

Note that if V = W ⊗ U and μ is the identity map, then Q is generated as an E-module by Q_{0}, so the complex F_{•}(μ) is irredundant by Proposition 2.2. The proof in Eisenbud-Popescu [1999] shows that F_{•}(μ) remains exact when μ is specialized to a certain pairing coming from the canonical module of a generic set of γ points in ℙ^{r} for suitable γ, r. Here we are instead interested in knowing whether F_{•}(μ) has an irredundant quotient complex with the same first and last terms as F_{•}(μ). This condition turns out to be a lot easier to analyze!

Following Eisenbud [1988] we say that μ : W ⊗U → V is 1-generic if μ(a⊗b) ≠ 0 for all nonzero vectors a ∈ W and b ∈ U. We say that μ is geometrically 1-generic if the induced pairing μ ⊗ \overline{k} is 1-generic, where \overline{k} is the algebraic closure of k. If μ is geometrically 1-generic, then by a classic and elementary result of Hopf the dimensions u, v, w of the three vector spaces satisfy the inequality v ≥ u + w − 1.

The main result of this section is the following:

Theorem 3.1 If the pairing μ : W ⊗ U → V is geometrically 1-generic, and W ≠ 0 and U ≠ 0, then F_{•}(μ) and its maximal irredundant quotient F′_{•}(μ) have the same last term F_{w−1} = F′_{w−1}, that is

F′_{•}(μ) : ∧^{w}W ⊗ D_{w−1}(U) ⊗ S(−w + 1)  F′_{w−2}  ⋯  F′_{1}  W ⊗ S.

Proof. We use notation as above. Let Q′ be the submodule of Q generated by Q_{0}, and set P = Q^{∗}, P′ = Q^{′∗}, so that F_{•} = L(P) and F′_{•} = L(P′). By Proposition 2.2, the conclusion of the Theorem is equivalent to the statement that the E-multiplication map

[figure: m :; E_{w−1} ⊗_{k} P_{0}; P_{−w+1}; ∧^{w−1}V^{∗} ⊗ W^{∗}; ‖; ∧^{w}W^{∗} ⊗ Sym_{w−1}(U^{∗}); ‖]

is surjective. Via the natural identification W^{∗} ⊗ ∧^{w}W ≅ ∧^{w−1}W this is equivalent to the map

m′ : ∧^{w−1}V^{∗} ⊗_{k} ∧^{w−1}W Sym_{w−1}(U^{∗})

induced by μ being surjective. A straightforward computation shows that the image of m′ is the space generated by the (w − 1) × (w − 1) minors of the linear map of free Sym(U^{∗})-modules

μ̄ : W ⊗ Sym(U^{∗}) V ⊗ Sym(U^{∗})(1)

associated to μ, and the next lemma shows that under the 1-genericity assumption these minors span Sym_{w−1}(U^{∗}). This proves the theorem.

Lemma 3.2 If μ : W ⊗ U → V is a geometrically 1-generic pairing of finitedimensional vector spaces, and if d is an integer such that d ≤ dim W, then the d × d minors of the associated linear map μ̄ : W ⊗ Sym(U^{∗}) → V ⊗ Sym(U^{∗})(1) of free Sym(U^{∗})-modules span Sym_{d}(U^{∗}).

Proof. It is enough to prove the lemma for an algebraically closed base field k, so that the Nullstellensatz holds. Also, u, v, w continue to be the dimensions of the three vector spaces.

We first show that, for an arbitrary subspace W′ ⊂ W of dimension d, the d×d minors of the restricted map

μ̄′ : W′ ⊗ Sym(U^{∗}) V ⊗ Sym(U^{∗})(1),

regarded as polynomial functions on U, have a common zero only at the origin 0 ∈ U. For μ̄′ is essentially a v × d matrix whose entries are linear functions on U, so its d × d minors have a common zero at a point b ∈ U if and only if the matrix’s columns, when evaluated at b, are not linearly independent. But this means that there is a 0 ≠ a ∈ W′ such that μ(a ⊗ b) = 0, which gives b = 0 by 1-genericity.

A general position argument shows that, for a general projection V [figure] V′ with dim V′ = d + u − 1, the only common zero of the d × d minors of the composite map

μ̄′′ : W′ ⊗ Sym(U^{∗}) V′ ⊗ Sym(U^{∗})(1),

is still 0 ∈ U. (One may argue that the dual of μ̄′ may be regarded as a surjective map O^{v} [figure] O^{d}(1) of vector bundles on ℙ^{u−1}, and a rank d vector bundle generated by global sections on a (u − 1) dimensional space can always be generated by just d + u − 1 of them; see for example Eisenbud-Evans [1973], or Serre [1958]).

The expected codimension of the locus defined by the d × d minors of the (d+u−1)×d matrix representing μ̄′′ is u, which is equal to the actual codimension. Thus the Eagon-Northcott complex resolving the minors of μ̄′′ is exact, and it follows that there are \binom{d+u−1}{d} linearly independent such minors. Since this is also the dimension of Sym_{d}(U^{∗}), we see that the minors span Sym_{d}(U^{∗}) as required.

We say that a map of complexes α_{•} : F_{•} → G_{•} is a degreewise split injection if each α_{i} : F_{i} → G_{i} is a split injection (this is weaker than being a split injection of complexes); in this case we say that F_{•} is a degreewise direct summand of G_{•}.

The irredundant complexes F′_{•}(μ) arise in geometric situations as follows.

Theorem 3.3 Let L, L′ be two line bundles on a scheme X over a field k, and let L′′ := L ⊗ L′. Let W ⊂ H^{0}(L), U ⊂ H^{0}(L′) be nonzero finite-dimensional linear series such that the multiplication

μ : W ⊗ U H^{0}(L′′)

is geometrically 1-generic. Let V ⊂ H^{0}(L′′) be a finite-dimensional linear series containing W · U. Let S := Sym(V), and let M ⊂ ⊕_{n≥0}H^{0}(L ⊗ L′′^{⊗n}) be a finitely generated graded S-submodule such that W ⊂ M_{0}. Then the maximal irredundant quotient

F′_{•}(μ) : ∧^{w}W ⊗ D_{w−1}(U) ⊗ S(−w + 1)  F′_{w−2}  ⋯  F′_{1}  W ⊗ S

of the linear free complex F_{•}(μ) injects as a degreewise direct summand of the minimal free resolution of M. (Here as above w = dim W.)

Proof. Let G_{•} be the minimal free resolution of M over S. The hypotheses on M imply that G_{0} = (W ⊗ S) ⊕ L, with L a free S-module. We can construct inductively a morphism α_{•} : F_{•}(μ) → G_{•}, such that \overline{α_{0}} : F_{0}(μ)/𝔪F_{0}(μ) → G_{0}/𝔪G_{0} is a monomorphism, as a lifting

[figure: ⋯; ∧^{3}W ⊗ D_{2}U ⊗ S(−2); ∧^{2}W ⊗ U ⊗ S(−1); W ⊗ S; ⋯; G_{2}; . . . . . . . .; G_{1}; α_{1}; . . . . . . . .; (W ⊗ S) ⊕ L; ∩; M]

For instance, α_{1} exists since the composition ∧^{2}W ⊗ U ⊗ S(−1) → W ⊗ S → M vanishes. So by Lemma 2.3, α_{•} induces a degreewise split inclusion F′_{•}(μ) ^{⊂} G_{•}, with F′_{•}(μ) having the form given in Theorem 3.1 (since μ is geometrically 1-generic).

It is easy to see that if X is a geometrically reduced and irreducible scheme, then any pairing μ : W ⊗ U V induced by multiplication of sections as in Theorem 3.3 is geometrically 1-generic. The hypothesis sometimes holds for reducible or nonreduced schemes too, as in the case we will use for the proof of Theorem 1.1.

The next example shows that the complexes F_{•}(μ) and F′_{•}(μ) are not always the same.

Example 3.4 If we take U = W = H^{0}(O_{ℙ^{1}}(2)) with μ the multiplication map to V = H^{0}(O_{ℙ^{1}}(4)), that is

μ : k[s, t]_{2} ⊗ k[s, t]_{2} = W ⊗ U V = k[s, t]_{4},

then μ is geometrically 1-generic because k[s, t] is a domain and remains so over the algebraic closure of k. The module M = ⊕_{n≥0}H^{0}(O_{ℙ^{1}}(2)⊗O_{ℙ^{1}}(4n)) of Theorem 3.3 is the graded module associated to the line bundle O_{ℙ^{1}}(2 points) on ℙ^{1}, regarded as a module over the homogeneous coordinate ring S of ℙ^{4} via the embedding of ℙ^{1} as the rational normal quartic C. This line bundle is ω_{C}(1), the twist of the canonical line bundle by the hyperplane line bundle, so the minimal resolution of M is the dual of the minimal resolution of the homogeneous coordinate ring of C, suitably shifted. It has the form

G_{•} : 0  S(−4)  S^{6}(−2)  S^{8}(−1)  S^{3}  M  0.

On the other hand, F_{•} has the form

F_{•}(μ) : 0  S^{6}(−2)  (S^{3} ⊗ S^{3})(−1) ^{δ_{1}(μ)} S^{3},

and F_{1} = S^{9}(−1) cannot inject into G_{1} = S^{8}(−1). The columns of the matrix of δ_{1}(μ) have exactly one k-linear dependence relation, so the maximal irredundant quotient of F_{•} has the form

F′_{•} : 0  S^{6}(−2)  S^{8}(−1)  S^{3},

so in particular F′_{2} = F_{2}, in accordance with Theorem 3.1. By Lemma 2.3 any map from F_{•} to G_{•} lifting the identity on F_{0} is a degreewise split monomorphism on F′_{•}; in fact F′_{•} is isomorphic to the linear strand of G_{•} in this case.

4 Failure of the Minimal Resolution Conjecture

Before proving Theorem 1.1, we present the bare bones of Gale duality. The reader who wishes to go further into this rich and beautiful subject can consult Eisenbud-Popescu [1999, 2000], or Dolgachev-Ortland [1988].

Let Γ ⊂ ℙ^{r} be a set of γ = r+s+2 points. The linear forms on ℙ^{r} define a linear series V ⊂ H^{0}(O_{Γ}(1)). The orthogonal complement V^{⊥} ⊂ H^{0}(O_{Γ}(1))^{∗} is identified by Serre duality with a linear series (ω_{Γ})_{−1} ⊂ H^{0}(K_{Γ}(−1)) (where K_{Γ} denotes the canonical sheaf of Γ). Under mild hypotheses on Γ, the linear series (ω_{Γ})_{−1} is very ample and gives an embedding whose image Γ′ ⊂ ℙ^{s} is the Gale transform of Γ. As a set it is well-defined only modulo linear changes of coordinates on ℙ^{s} and ℙ^{r}.

Concretely, if the columns of the (r + 1) × γ matrix M represent the points of Γ ⊂ ℙ^{r}, and if N is a γ × (s + 1) matrix whose columns are a basis of ker(M), then the rows of N represent the points of Γ′ ⊂ ℙ^{s}.

In the proof of Theorem 1.1 we need four facts about the Gale transform, which the reader may easily verify (or find in Eisenbud-Popescu [1999, 2000]).

– The subset Γ′ ⊂ ℙ^{s} is defined for a general Γ. In fact, it is defined as long as no subset containing all but two of the points of Γ lies in a hyperplane of ℙ^{r}.

– The Gale transform of Γ′ is Γ.

– There is a natural identification of V = H^{0}(O_{ℙ^{r}}(1)) with (ω_{Γ′})_{−1}.

– Γ is general if and only if Γ′ is general. More formally, a GL(s + 1)-invariant Zariski open condition on Γ′ induces a GL(r+1)-invariant Zariski open condition on Γ.

Proof of Theorem 1.1. Let s ≥ 3 be an integer and set r = \binom{s+1}{2} + δ. Suppose that

0 ≤ δ ≤ \binom{s}{2} − \{ \begin{matrix}1 & if s ≤ 4, \\ 2 & if s ≥ 5, \end{matrix}

and let Γ ⊂ ℙ^{r} be a set of γ(r, s) = r + s + 2 points, in linearly general position. We shall show that the Minimal Resolution Conjecture fails for Γ. From this statement it is easy to solve for γ in terms of r, obtaining the range given in Theorem 1.1.

Let I_{Γ} be the homogeneous ideal of Γ ⊂ ℙ^{r} and let

ω_{Γ} = Ext_{S}^{r−1}(I_{Γ}, S(−r − 1))

be the canonical module. The free resolution of ω_{Γ} is, up to a shift in degree, the dual of the resolution of S/I_{Γ}.

We will assume that Γ is a general set of points such that: its Gale transform Γ′ ⊂ ℙ^{s} is defined and is in linearly general position, Γ imposes independent conditions on quadrics, and Γ′ is not contained in any quadric (note that \binom{s+2}{2} < γ(r, s) ≤ \binom{r+2}{2}).

Since γ(r, s) > \binom{s+2}{2}, it follows that Γ′ does not impose independent conditions on quadrics of ℙ^{s}, and thus H^{1}(I_{Γ′}(2)) ≠ 0. We set

U := (ω_{Γ′})_{−2} = Hom_{k}(H^{1}(I_{Γ′}(2)), k) ≠ 0.

(In this setting we have dim U = δ + 1.) Write W = H^{0}(O_{ℙ^{s}}(1)), and V = H^{0}(O_{ℙ^{r}}(1)). Multiplication induces a natural pairing

μ : W ⊗ U V = (ω_{Γ′})_{−1}.

We now carry out the checks needed to apply Theorem 3.3. First, Γ′ ⊂ ℙ^{s} is in linearly general position, so by Kreuzer [1994] the pairing μ is 1-generic over any field and thus geometrically 1-generic. Second, Γ ⊂ ℙ^{r} imposes independent conditions on quadrics, so (ω_{Γ})_{−d} = 0 for d ≥ 2. Hence ω_{Γ}(−1) is a finitely generated graded S-submodule of ⊕_{n≥0}H^{0}(K_{Γ}(n − 1)) containing W = (ω_{Γ})_{−1} in degree 0.

Therefore the irredundant linear free quotient F′_{•}(μ) of the complex F_{•}(μ) injects onto a degreewise split direct summand of the minimal free resolution of ω_{Γ}(−1), which is the dual of the minimal free resolution of (S/I_{Γ})(r + 2). Hence the true graded betti number β_{(r−s),(r−s+2)} for S/I_{Γ} satisfies

β_{(r−s),(r−s+2)} ≥ rank F′_{s}(μ) = \binom{s + δ}{δ} > 0.

The corresponding expected graded betti number β̃_{(r−s),(r−s+2)} is found in Eisenbud-Popescu [1999], or it can be computed by taking the Hilbert series for Γ

∑_{i} dim(S/I_{Γ})_{i} · t^{i} = 1 + (r + 1)t + γ(r, s) · \frac{t^{2}}{1 − t} = \frac{∑_{j} b_{j}t^{j}}{(1 − t)^{r+1}}

and calculating that

(−1)^{r−s}b_{r−s+2} = \frac{(2δ + 4 − s^{2} + s)}{(s^{2} − s + 2δ + 4)} · \binom{\binom{s+1}{2} + δ}{s}.

This number is the alternating sum of the graded betti numbers of degree r − s + 2, in this case β_{(r−s),(r−s+2)} − β_{(r−s+1),(r−s+2)}. The expected graded betti number is thus

β̃_{(r−s),(r−s+2)} = max ((−1)^{r−s}b_{r−s+2}, 0) .

The MRC certainly fails when β̃_{(r−s),(r−s+2)} = 0, which is equivalent to 0 ≤ δ ≤ \binom{s}{2}−2. The MRC also fails for s = 3, 4 and δ = \binom{s}{2}−1 because \binom{s+δ}{δ} > β̃_{(r−s),(r−s+2)} in those cases. (For s = 4 this is Boij’s example!) Straightforward computation shows that there are no other cases where \binom{s+δ}{δ} > β̃_{(r−s),(r−s+2)}.

Schreyer observed around 1983 (unpublished) that the minimal resolution conjecture fails for linearly normal curves of degree d > g^{2} − g, g ≥ 4 A result of Green and Lazarsfeld [1988] implies that this bound can be improved to d > 3g − 3. We can use the technique developed above to give a new proof of the relevant part of the Green-Lazarsfeld result, recovering the failure of the MRC in a new way.

Theorem 4.1 Let C be a smooth curve of genus g ≥ 2, let L ∈ Pic^{d}(C) be a line bundle on C of degree d ≥ 2g + 2, and denote by φ_{|L|} : C → ℙ^{d−g} the embdding defined by the complete linear system |L|.

a) If H^{0}(L ⊗ ω_{C}^{−1}) ≠ 0 (that is, equivalently, if the curve φ_{|L|}(C) ⊂ ℙ^{d−g} has a (d−2g +1)-secant ℙ^{d−2g−1}), then the maximal irredundant quotient of the free linear complex

F_{•} := L(⊕_{l≥0} ∧^{l+1} H^{0}(ω_{C}) ⊗ D_{l}(H^{0}(L ⊗ ω_{C}^{−1}))

injects as a degreewise direct summand of the minimal free resolution of the Sym(H^{0}(L))-module ⊕_{n≥0}H^{0}(ω_{C} ⊗ L^{n}). In particular φ_{|L|}(C) ⊂ ℙ^{d−g} doesn’t satisfy the property N_{d−2g}.

b) If g ≥ 4 and d > 3g − 3, then the Minimal Resolution Conjecture fails for φ_{|L|}(C) ⊂ ℙ^{d−g}.

Remark 4.2 By Green [1984], φ_{|L|}(C) ⊂ ℙ^{d−g} satisfies the property N_{d−2g−1} (that is the embedding is projectively normal, the homogeneous ideal I_{φ_{|L|}(C)} is generated by quadrics and all syzygies in the first (d − 2g − 2) steps in its minimal free resolution are linear). On the other hand, if C is non-hyperelliptic then property N_{d−2g} fails for φ_{|L|}(C) ⊂ ℙ^{d−g} iff the curve has a (d − 2g + 1)-secant (d − 2g − 1)- plane, by Green-Lazarsfeld [1988, Theorem 2]. Theorem 4.1 a) spells out explicitly this failure!

Proof of Theorem 4.1: By hypothesis d ≥ 2g + 2 so φ_{|L|}(C) ⊂ ℙ^{d−g} is arithmetically Cohen-Macaulay, its homogeneous ideal I_{φ_{|L|}(C)} is generated by quadrics and is 2-regular. Now Theorem 3.3, applied to the geometrically 1-generic pairing

μ : H^{0}(ω_{C}) ⊗ H^{0}(L ⊗ ω_{C}^{−1}) → H^{0}(L)

yields the complex F_{•} = F_{•}(μ) whose maximal iredundant quotient injects as a degreewise direct summand of the minimal free resolution of ⊕_{n≥0}H^{0}(ω_{C} ⊗ L^{n}), which is the dual of the minimal free resolution of S/I_{φ_{|L|}(C)}(d − g + 1). Here and in the sequel S = Sym(H^{0}(L)). It follows that the graded betti number β_{d−2g,d−2g+2} of S/I_{φ_{|L|}(C)} satisfies the inequality

β_{d−2g,d−2g+2} ≥ dim Sym_{g−1}(H^{0}(L ⊗ ω_{C}^{−1})) = \binom{d − 2g + 1}{g − 1}.

In particular β_{d−2g,d−2g+2} ≥ 1, and so φ_{|L|}(C) ⊂ ℙ^{d−g} doesn’t satisfy property N_{d−2g}, which finishes the proof of part a).

Since the embedding φ_{|L|}(C) ⊂ ℙ^{d−g} is arithmetically Cohen-Macaulay, its hyperplane section Γ ⊂ ℙ^{d−g−1} has the same graded Betti numbers as φ_{|L|}(C). Thus, as in Eisenbud-Popescu [1999] or the proof of Theorem 1.1 above, we obtain for the alternated sum of graded betti numbers of degree d − 2g + 2 for S/I_{φ_{|L|}(C)}:

β_{d−2g,d−2g+2} − β_{d−2g+1,d−2g+2} = \frac{(d − g^{2} + g)}{(d − 2g + 2)}\binom{d − g − 1}{g − 1}.

If 3g − 2 ≤ d ≤ g^{2} − g the Minimal Resolution Conjecture predicts that β_{d−2g,d−2g+2} = 0, while from a) we get β_{d−2g,d−2g+2} ≥ 1; hence the Minimal Resolution Conjecture fails for d in the above range.

For d ≥ g^{2} − g + 1, the expected β_{d−2g+1,d−2g+2} is zero while the expected β_{d−2g,d−2g+2} is always larger than \binom{d−2g+1}{g−1} and so the linear complex in a) doesn’t account for the failure of the MRC.

For g ≥ 4, there exist a g_{g−1}^{1} = |O_{C}(D)| on C. The pairing

η : H^{0}(O_{C}(D)) ⊗ H^{0}(L ⊗ O_{C}(−D)) → H^{0}(L)

is 1-generic and thus the pairing η defines a 2×(d−2g+2) matrix with linear entries in ℙ^{d−g} whose 2×2 minors vanish in the expected codimension (see Eisenbud [1988]). Therefore the Eagon-Northcott complex resolving the 2×2-minors of this matrix is a linear exact complex of length d−2g+1 which injects as a degreewise direct summand in the top strand of the resolution of I_{φ_{|L|}(C)}. In particular β_{d−2g+1,d−2g+2} ≥ (d − 2g + 1) > 1 so the Minimal Resolution Conjecture fails for φ_{|L|}(C) also when d ≥ g^{2} − g + 1. This concludes the proof of part b).

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Author Addresses: David Eisenbud Department of Mathematics, University of California, Berkeley, Berkeley CA 94720

de@msri.org

Sorin Popescu Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, and Department of Mathematics, Columbia University, New York, NY 10027

sorin@math.sunysb.edu

Frank-Olaf Schreyer FB Mathematik, Universität Bayreuth, D-95440 Bayreuth, Germany

schreyer@btm8x5.mat.uni-bayreuth.de

Charles Walter Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS) Université de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France

walter@math.unice.fr