ON GRADED BETTI NUMBERS FOR FINITE SUBSETS OF CURVES

SERGE LVOVSKI

1. Introduction

When a computer graphics program draws a curve on the screen, it replaces the curve by its finite subset, and this subset serves as a good enough approximation. The present paper grew out of an attempt to make this idea of approximation precise: what could an algebraic geometer mean by saying that curves are approximated by their finite subsets?

To be more specific, consider a projective curve C ⊂ ℙ^{n} and various finite subsets X ⊂ C. Let us compare the defining equations of C and X; do the equations of X “converge” to those of C as the subset X ⊂ C grows? Actually we work not only with the equations, but with relations among these equations, relations among these relations … – in short, we compare minimal resolutions for homogeneous coordinate rings of X and C and try to learn what happens as new points of C are added to the set X. It is not hard to show that minimal resolutions of the sets X actually approximate the minimal resolution of C, and that a weakened form of this assertion is true for varieties of arbitrary dimension – cf. Theorem 1.1 below. However, computer experiments with Macaulay [2] suggest that something much more interesting happens as the subset X grows: eventually the minimal resolution for X splits into two parts, one of which is just the minimal resolution for C, while the other behaves very regularly. This suggests Conjecture 1.2 (see below), which we call “Periodicity conjecture”.

I was not able to prove this conjecture. Instead, I provide some evidence for it by proving some of its consequences. I hope that the techniques we use in section 2, as well as some of the proved results, are of independent interest.

Before stating the results and conjectures, let us fix some notation. Assume that the ground field k is of characteristic 0, algebraically

closed, and uncountable.

If ℙ^{n} = Proj R, where R = k[T_{0}, …, T_{n}], and if X ⊂ ℙ^{n} is a closed subscheme, we denote the homogeneous ideal of X by I_{X} ⊂ R and put R_{X} = R/I_{X}. The minimal free graded resolution of the R-module R_{X} will be denoted by L_{·}(X). Thus, L_{0}(X) = R and L_{m}(X) = 0 for m > n. If L_{i} = ⨁_{j} R(−i − j)^{b_{ij}(X)}, then the numbers b_{ij}(X) are called graded Betti numbers of X. Our notation for graded Betti numbers agrees with that adopted by the program Macaulay but differs from, for example, Birkenhake’s [3]

Now we can state the approximation result (see Definition 3.4 below for the precise meaning of the words “generic infinite sequence of points”):

Theorem 1.1 (Approximation). Suppose that V ⊂ ℙ^{n} is an irreducible projective variety and {P_{t}}_{t∈ℕ} is a generic sequence of distinct points of V; put X_{i} = { P_{1}, …, P_{i} } ⊂ V. Then for any j ≥ 0 there exists an integer N > 0 such that b_{rs}(X_{i}) = b_{rs}(V) whenever s ≤ j and i ≥ N.

If dim V = 1, then this property holds for any sequence of distinct points P_{i} ∈ V and for any algebraically closed field of characteristic 0.

If the ground field is countable and dim V > 1, this theorem asserts nothing since in this case any property of infinite sequences is true for a generic sequence.

In the language of Betti diagrams (cf. [5]; by definition, Betti diagram of a closed subscheme X ⊂ ℙ^{n} is a rectangular table with b_{ij}(X) at the intersection of the i-th column and the j-th row; rows are numbered from top to bottom and columns from left to right), this means that, for all n ≫ 0, Betti diagram of X_{n} looks as follows: in the first few rows we see the Betti diagram for the curve C, an array of empty rows follows, and then a “tail”, which distinguishes the Betti diagram for X_{n} from that of C, appears. The following conjecture asserts that the “tail” behaves very regularly:

Conjecture 1.2 (Periodicity). Suppose that C ⊂ ℙ^{n} is an irreducible projective curve of degree d and that P = {P_{m}}_{m∈ℕ} is a generic sequence of distinct points of C. Put δ = max{ j | ∃i: b_{ij}(C) ≠ 0 }; for each m ∈ ℕ put X_{m} = { P_{1}, …, P_{m} } ⊂ C and t_{m} = min{ j > δ | ∃i: b_{ij}(X_{m}) ≠ 0 }. Then for all m ≫ 0 we have: t_{m+d} = t_{m} + 1, and if j ≥ t_{m}, then b_{i,j+1}(X_{m+d}) = b_{ij}(X_{m}) for all i.

In the language of Betti diagrams this means that, for all n ≫ 0, the “tail” repeats periodically with period d = deg C; after every d steps, the tail shifts one row down, leaving zeroes in the vacant places (see the appendix for an illustration).

Further Macaulay experiments suggest that even more may be true: periodicity takes place if we add points of C to an arbitrary closed Y ⊂ ℙ^{n} such that C ⊄ Y.

Conjecture 1.3 (Periodicity, strong form). Suppose that Y ⊂ ℙ^{n} is a closed subset and C ⊂ ℙ^{n} is a closed curve not contained in Y. Put d = deg C and δ = max{ j | ∃i: b_{ij}(Y ∪ C) ≠ 0 }; for each m ≥ 0, put X_{m} = Y ∪ { P_{1}, …, P_{m} } and t_{m} = min{ j > δ | ∃i: b_{ij}(Y ∪ X_{m}) ≠ 0 }. Then for all m ≫ 0 we have t_{m+d} = t_{m} + 1, and if j ≥ t_{n}, then b_{i,j+1}(X_{m+d}) = b_{ij}(X_{m}) for all i. Moreover, the periodic pattern appearing for large m in Betti diagrams of Y ∪ {P_{1}, …, P_{m}}) depends on C, but not on Y.

Presently there is no evidence for this conjecture besides computer experiments.

The paper is organized as follows. In section 2 we explain how to express Koszul cohomology of an embedded projective variety via its Macaulay’s inverse system. This allows us to prove the following result, which is used in subsequent computations (the ground field is algebraically closed of characteristic 0): Proposition 1.4. Let X ⊂ V ⊂ ℙ^{n} be Zariski closed sets. Suppose that, for an integer m > 0, linear spans of the m-th Veronese images of X and V coincide. Then b_{rs}(X) = b_{rs}(V) whenever s < m, and b_{rm}(X) ≥ b_{rm}(V) for all m.

This proposition means that the syzygies of degree < m are completely determined by equations of degree ≤ m.

In section 3 we make precise the notion of a generic infinite sequence and prove the approximation property (Theorem 1.1). Section 4 contains various auxiliary results on graded Betti numbers. Most of them must be known to the experts. In particular, we compute graded Betti numbers for subvarieties of minimal degree. Our method allows us to do it almost for free, without using the determinantal techniques (cf. [1, II.5.3]).

In section 5 we prove two consequences of Conjecture 1.2. First, we show that the lower right corners of Betti diagrams of subsets X_{n} ⊂ C behave as predicted by the conjecture. This is our Proposition 5.1. Then we prove Conjecture 1.2 for rational normal curves. It turns out that in this case the assertion of Conjecture 1.2 is true for arbitrary sequences without repetitions – this is our Proposition 5.7.

Finally, in section 6 we compute graded Betti numbers for some finite subsets X ⊂ C, where C ⊂ ℙ^{n} is a rational normal curve of degree n. Proposition 5.7 says that graded Betti numbers for finite subsets X ⊂ C depend only on card(X), and we find explicit formulas for the cases card(X) ≡ 0, 1 or 2 (mod n).

Acknowledgements. I am greatly indebted to F.L. Zak for his attention to this work, helpful criticism, and numerous stimulating discussions. I would like to thank L. Bădescu, B. Karpov, S. Kuleshov, A. Shen, and A.N. Tyurin for useful discussions. Last but not least, I would like to express my gratitude to Max-Planck-Institut für Mathematik (Bonn) for hospitality and providing excellent working conditions.

Notation and conventions

The ground field k will be always assumed to have characteristic 0. ℙ(E) means Proj Sym(E^{∗}); thus, over an algebraically closed field closed points of ℙ(E) correspond to lines in (fibers of) E. If X ⊂ ℙ^{n} = ℙ(E) is a closed subset, then X̂ ⊂ E denotes the affine cone over X and ⟨X⟩ ⊂ ℙ^{n} denotes the linear span of X. The sheaf on ℙ^{n} corresponding to a graded module M will be denoted by M̃.

For any multi-index α = (α_{0}, …, α_{n}), put

|α| = ∑ α_{i},  α! = ∏ α_{i}!,
X^{α} = ∏ X_{i}^{α_{i}},  D_{α} = ∏ (∂/∂X_{i})^{α_{i}} .

For any number t, we write ⌊t⌋ for max{ m ∈ ℤ | m < t}.

If X ⊂ ℙ^{n} is a closed subscheme, we will say that X is projectively normal if H^{1}(ℙ^{n}, I_{X}(m)) = 0 for all m ∈ ℤ. This is equivalent to saying that depth(R_{X}) ≥ 2. If X is irreducible and reduced, its projective normality is equivalent to linear normality of Veronese images v_{m}(X) for all m ≥ 0, and finite sets containing more than one point are never projectively normal.

Finally, let us fix our notation with Veronese maps. If (x_{0} : ⋯: x_{n}) are homogeneous coordinates in ℙ^{n}, consider a projective space ℙ^{\binom{n+m}{n}−1} in which homogeneous coordinates y_{α} are numbered by multi-indices α = (α_{0}, …, α_{n}) such that |α| = m. By Veronese map v_{m}: ℙ^{n} → ℙ^{\binom{n+m}{n}−1} we mean the map defined by the formulas y_{α} = x^{α}/α! . This map differs from the usual Veronese nap by a projective automorphism of ℙ^{\binom{n+m}{n}−1}, which does not affect geometric properties of the map.

2. Inverse systems and Koszul cohomology

Let R = ∑_{i=0}^{∞} R_{i} be the graded ring of polynomials in T_{0}, …, T_{n}; denote by S = ∑_{i=0}^{∞} S_{i} the graded ring of polynomials in ξ_{0}, …, ξ_{n}. If we let T_{j} act on S by the rule T_{j} · φ ↦ ∂φ/∂ξ_{j}, S becomes a graded R-module.

It is well known (cf. [4, §1, ex. 32b]) that S is injective as an Rmodule; moreover, it is an injective hull of the R-module k.

Elements of the R-module S are known as Macaulay’s inverse polynomials (cf. [6]).

The following definition is due to Macaulay.

Definition 2.1. A homogeneous linear subspace J ⊆ S which is an R-submodule with respect to the above action of R will be called an inverse system in S.

For any j ≥ 0, there is a pairing S_{j} ⊗ R_{j} → k which acts by the formula

⟨φ, f⟩ ↦ φ(∂/∂T_{0}, …, ∂/∂T_{n}) · f. (2.1)

Proposition 2.2. Pairing (2.1) is perfect.

Proposition 2.3. If φ ∈ S_{m}, f ∈ R_{m−1}, and j ∈ [0; n], then

⟨∂φ/∂ξ_{j}, f⟩ = ⟨φ, T_{j}f⟩; (2.2)

if φ ∈ S_{m−1}, f ∈ R_{m}, and j ∈ [0; n], then

⟨φ, ∂f/∂T_{j}⟩ = ⟨ξ_{j}φ, f⟩ . (2.3)

Proof. To prove (2.2) it suffices to assume that φ = ξ_{j}^{t} · ψ, where ψ does not depend on ξ_{j}. Put g = ψ (∂/∂T_{0}, …, ∂/∂T_{n}) · f. Then the right-hand side of (2.2) equals (∂/∂ξ^{j})^{t} · (T_{j}g), which by Leibnitz rule equals T_{j} (∂/∂ξ^{j})^{t} · g + j (∂/∂ξ^{j})^{t−1} · g. Since deg g = t − 1 < m, the first summand is zero, whence the desired equality. The identity (2.3) is obvious.

Definition 2.4. If I ⊂ R is a homogeneous linear subspace, put I^{⊥} = ⨁_{j} ann(I_{j}) ⊂ S. If J ⊂ S is a homogeneous linear subspace, put J^{⊥} = ⨁_{t} ann(J_{t}) ⊂ R (in both cases ann means “annihilator with respect to the pairing (2.1)”).

We will say that I^{⊥} is the annihilator of I and J^{⊥} is the annihilator of J.

If I ⊂ R is a homogeneous linear subspace, then I is an ideal in R iff I^{⊥} is an inverse system in S. The correspondence I ↦ I^{⊥} is an inclusion-reverting 1-1 correspondence between homogeneous ideals of R and inverse systems of S.

Proposition 2.5. If homogeneous polynomials F_{1}, …, F_{r} generate an ideal I ⊂ R and I^{⊥} ⊂ S is the annihilator of I, then each homogeneous component (I^{⊥})_{m} ⊂ S_{m} consists of all Φ ∈ S_{m} such that

F_{i}(∂/∂ξ_{0}, …, ∂/∂ξ_{n})(Φ) = 0 (2.4)

for all i ∈ [1; r].

Proof. Put J = I^{⊥} and suppose that Φ ∈ J_{m}. Then ⟨Φ, MF_{j}⟩ = 0 for any j ∈ [1; r] and any M ∈ R_{m−deg F_{j}}. If Φ′ = F_{j}(∂/∂ξ_{0}, …, ∂/∂ξ_{n})Φ, then (2.2) implies that M(∂/∂ξ_{0}, …, ∂/∂ξ_{n})Φ′ = 0 for any M ∈ R_{m−deg F_{j}}, whence Φ′ = 0. The inverse inclusion is obvious.

From now on we will consider the projective space ℙ^{n} as Proj R. Pairing (2.1) identifies S_{1} with (R_{1})^{∗}; thus, we may (and from now on will) identify ℙ^{n} with ℙ(S_{1}). If v = ∑ c_{i}ξ_{i} ∈ S_{1}, f ∈ R_{m}, then by f(v) we mean f(ξ_{0}, …, ξ_{n}). If k is algebraically closed and X is reduced, then v ∈ X̂ iff f(v) = 0 for all homogeneous f ∈ I_{X}.

Definition 2.6. If X ⊂ ℙ^{n} is a closed subscheme and I_{X} ⊂ R is its graded ideal, then the inverse system (I_{X})^{⊥} ⊂ S will be denoted by J_{X} and called the inverse system of X.

Proposition 2.7. If k is algebraically closed and X ⊂ ℙ^{n} is reduced, then (J_{X})_{r} is linearly generated by the polynomials φ_{x} = ∑_{|α|=r}(x^{α}/α!)ξ^{α} for all x ∈ X̂.

Proof. Take (…, T^{α}, …)_{|α|=r} as a basis of R_{r}; then (…, ξ^{α}/α!, …) will be the dual basis of S_{r}. If f = ∑_{|α|=r} c_{α}T^{α} ∈ R_{r}, then f ∈ I_{r} iff ∑_{|α|=r} c_{α}x^{α} = 0 for all x ∈ X̂ iff ⟨φ_{x}, f⟩ = 0 for all x ∈ X̂. Thus, ((I_{X})_{r})^{⊥} is generated by the φ_{x}’s with x ∈ X̂.

Corollary 2.8. If k is algebraically closed and p ∈ ℙ^{n} is a (closed) point that corresponds to a vector v ∈ X̂\0 ⊂ S_{1}\0, then J_{p} = k[v] ⊂ S.

Proposition 2.9. In the above setting, suppose that X ⊂ ℙ^{n} = ℙ(S_{1}) is a closed subset and p ∈ ℙ^{n} is the point corresponding to a vector v ∈ S_{1}. Then p ∈ X iff v^{r} ∈ J_{X} for all r > 0.

Proof. p ∈ X iff J_{p} ⊂ J_{X}; now Corollary 2.8 applies.

Corollary 2.10. If the ground field is algebraically closed and X ⊂ ℙ^{n} is a closed subset, then ℙ((J_{X})_{r})) ⊂ ℙ(S_{r}) coincides with ⟨v_{r}(X)⟩.

Corollary 2.11. Suppose again that k is algebraically closed. Let X ⊂ ℙ^{n} be a closed subset such that X = ⋃_{α} X_{α}, where all the X_{α}’s are closed in ℙ^{n}. Then J_{X} = ∑_{α} J_{X_{α}}.

If M = ⨁_{i} M_{i} is a graded R-module, one may consider a graded R-module M^{∗} = ∑_{i}(M_{−i})^{∗}, where the action T_{j}: (M^{∗})_{−i−1} → (M^{∗})_{−i} is the adjoint to T_{j}: M_{i} → M_{i+1}. In particular, R^{∗} = ⨁_{i≤0} S_{−i}, and if I ⊂ R is a homogeneous ideal, then (R/I)^{∗} = ⨁_{i≤0}(I^{⊥})_{−i}. Abusing the language, we will nevertheless write (R/I)^{∗} = I^{⊥}; this should not lead to confusion. If we put k = R/(T_{0}, …, T_{n}), then k^{∗} = k; for any finitely generated graded R-module M, the graded modules Tor_{j}^{R}(k, M) and (Ext_{R}^{j}(k, M^{∗}))^{∗} are naturally isomorphic.

Suppose that J ⊂ S is an inverse system. Then Ω^{r}(J) will denote the vector space of r-differential forms whose coefficients are polynomials in ξ_{0}, …, ξ_{n} lying in J. The spaces Ω^{r}(J) are naturally graded: namely, Ω^{r}(J) = ⨁ Ω_{s}^{r}(J), where Ω_{s}^{r}(J) consists of forms whose coefficients are homogeneous of degree s. If J = J_{X}, where X ⊂ ℙ^{n} is a closed subset (or subscheme), we will write Ω_{s}^{r}(X) instead of Ω_{s}^{r}(J); this should not lead to confusion.

Exterior differentiation maps Ω_{s}^{r}(J) into Ω_{s−1}^{r+1}(J). Thus, for an arbitrary integer l, exterior differentiation d is a graded homomorphism from Ω^{r}(J)(l) to Ω^{r+1}(J)(l + 1). Consider a graded complex

0 → Ω_{J}^{0} → Ω_{J}^{1}(1) → ⋯ → Ω_{J}^{n+1}(n + 1) → 0. (2.5)

and denote it by M_{J}^{·}. Its cohomology groups are naturally graded. We will write H^{r}(M_{J}^{·}) = ⨁_{s} H_{s}^{r}(J); thus, H_{s}^{r}(J) is the vector space of closed r-forms whose coefficients lie in J_{s}, modulo exterior derivatives of forms with coefficients in J_{s+1}. If J = J_{X}, we will write M^{·}(X) and H_{s}^{r}(X) instead of M^{·}(J_{X}) and H_{s}^{r}(J_{X}).

Proposition 2.12. Suppose that I ⊂ R is a homogeneous ideal. Then (Tor_{m}^{R}(k, R/I)_{i})^{∗} ≅ H_{i−m}^{m}(I^{⊥}).

Proof. Put J = I^{⊥}. The left-hand side is isomorphic to Ext_{m}^{R}(k, J). If we compute this Ext with the aid of the standard Koszul resolution K_{·} for k, which is

0 → R(−n − 1)^{\binom{n+1}{n+1}} → R(−n)^{\binom{n+1}{n}} → ⋯ → R(−1)^{\binom{n+1}{1}} → R ↠ k,

then Hom(K_{·}, J) coincides with the complex (2.5), whence the conclusion.

Remark 2.13. Putting I = 0 in the above proposition, we see that Poincaré lemma for forms with polynomial coefficients is essentially equivalent to injectivity of the module of inverse functions.

Corollary 2.14. If X ⊂ ℙ^{n} is a closed subscheme, then each graded Betti number b_{rs}(X) equals dim H_{s}^{r}(X).

Remark 2.15. For the first term of the resolution, i.e. for the minimal defining equations of a variety, the duality between Tor_{1}^{S}(k, R/I)_{r} and H_{r−1}^{1}(I^{⊥}) may be made explicit. To wit, suppose that I ⊂ R is a homogeneous ideal and that r ≥ 1 is an integer. Then [Tor_{1}^{R}(k, R/I)]_{r} ≅ I_{r}/R_{1}I_{r−1}. Let F_{1}, …, F_{s} ∈ I_{r} form a basis of this vector space. Since the pairing ⟨·, ·⟩ is perfect, there exist forms Φ_{1}, …, Φ_{s} ∈ S_{r} such that ⟨Φ_{i}, F ⟩ = 0 for any F ∈ R_{1}(I_{X})_{r−1} and ⟨Φ_{i}, F_{j}⟩ = δ_{ij}. Under the above assumptions, the differential forms ω_{i} = dΦ_{i} lie in Ω_{r−1}^{1}(I^{⊥}) and form a basis of H_{r−1}^{1}(I^{⊥}) that is dual to (F_{1}, …, F_{s}) with respect to the duality from Proposition 2.12.

Example 2.16. If X ⊂ ℙ^{n} is a quadric defined by the equation T_{0}^{2} + ⋯ + T_{n}^{2} = 0, then H_{r}^{m}(I_{X}^{⊥}) is not zero only if m = r = 0 or m = r = 1, and H_{1}^{1}(I_{X}^{⊥}) is generated by the cohomology class of any of the forms ξ_{i}dξ_{i}.

Example 2.17. A homogeneous ideal I is saturated, i. e. (I : 𝔪) = I, where 𝔪 ⊂ R is the maximal homogeneous ideal, iff depth(R/I) > 0; according to Proposition 2.12 and Ausalnder-Buchsbaum formula, this should be equivalent to vanishing of H_{·}^{n+1}(I^{⊥}), i.e. to the following assertion:

For any f ∈ I^{⊥}, there exist f_{0}, …, f_{n} ∈ I^{⊥} such that f =

∑_{i=0}^{n} ∂f_{i}/∂ξ_{i}.

This means that I^{⊥} = (ξ_{0}, …, ξ_{n})I^{⊥}, which is indeed just the dual of the condition (I : 𝔪) = I.

3. Approximation

Throughout this section we assume that the ground field is algebraically closed. We start with a very simple observation.

Proposition 3.1. Suppose that X ⊂ ℙ^{n} is a closed subset and that X = ⋃_{α} Y_{α∈A}, where Y_{α} is closed in ℙ^{n} for all α (the index set A may be infinite). For any finite B ⊂ A put Y_{B} = ⋃_{α∈B} Y_{α}. Then, for any r and m, H_{s}^{r}(X) = \underrightarrow{lim}_{B ⊂ Y, B is finite}H_{s}^{r}(J_{Y_{B}}).

Proof. Corollary 2.11 implies that J_{X} = ∑_{α∈A} J_{Y_{α}} = \underrightarrow{lim}_{B ⊂ Y, B is finite}J_{Y_{B}}. Since, for any inverse system J ⊂ S, H_{s}^{r}(J) is just the degree s component of Ext_{S}^{r}(k, J), and since the partially ordered set of finite subsets of Y is directed, Proposition 5 from [4, § 6, n^{0} 4] shows that

Ext_{S}^{r}(k, J_{X}) = \underrightarrow{lim}_{B} Ext_{S}^{r}(k, J_{Y_{B}}),

whence the conclusion.

If the family A consists of all (closed) points of X, the assertion of the above proposition may be made more precise. We begin with proving Proposition 1.4. It follows immediately from

Proposition 3.2. Let X ⊂ V ⊂ ℙ^{n} be Zariski closed subsets. Suppose that ⟨v_{m}(X)⟩ = ⟨v_{m}(V)⟩ for an integer m > 0. Then the natural homomorphisms H_{s}^{r}(X) → H_{s}^{r}(V) are isomorphisms for s < m and epimorphisms for s = m.

Proof. The condition ⟨v_{m}(X)⟩ = ⟨v_{m}(V)⟩ means that, for any hypersurface W of degree m such that W ⊃ X, we have W ⊃ V. If this condition holds, then it holds with m replaced by an arbitrary l < m as well. Hence, it follows from the hypothesis that ⟨v_{l}(X)⟩ = ⟨v_{l}(V)⟩ for any l ≤ m; Corollary 2.10 implies that Ω_{l}^{r}(X) = Ω_{l}^{r}(V) whenever l ≤ m. Since d(Ω_{·}^{·}(V) ∩ Ω_{l}^{r}(V)) = d(Ω_{l+1}^{r−1}(V)), the proposition is proved.

Corollary 3.3. If a closed subscheme X ⊂ ℙ^{n} is not contained in a hyperplane, then b_{i0}(X) = 0 for i > 0.

Proof. Put V = ℙ^{n} and m = 1 in the above proposition.

Now we can prove Theorem 1.1 for the case dim V = 1. Recall that in this case V ⊂ ℙ^{n} is an irreducible curve, P = {P_{m}}_{m∈ℕ} is a sequence of distinct points of the curve V, and we are to prove that for any j > 0 there exists a number N such that b_{rs}(V) = b_{rs}(X_{m}) whenever s ≤ j and m ≥ N, where X_{m} = { P_{1}, …, P_{m} }.

Proof. Any curve that is not a line can meet a proper linear subspace only in a finite number of points. Hence, for any j > 1 there exists an N such that ⟨X_{m}⟩ = ⟨v_{j+1}(V)⟩ whenever m ≥ N. Now Proposition 3.2 applies.

If dim V > 1, the approximation property can not hold for all sequences {P_{j}}. For example, all the P_{i}’s may lie on a proper subvariety of V, in which case the X_{i}’s cannot know about graded Betti numbers of V. However, this property holds for generic sequences of points.

Let us give a precise definition of “genericity” for infinite sequences. Suppose that V is an irreducible quasiprojective variety of positive dimension, and let V denote the set of all sequences p = {P_{i} ∈ V }_{i∈ℕ}.

Identifying V with V^{ℕ}, we may consider a natural projection p_{S}: V^{ℕ} → V^{S}, where S ⊂ ℕ is a finite subset (if S = { n_{1}, …, n_{l}}, then V^{S} is just the product of l copies of the variety V and p_{S} maps the sequence {P_{i}} to the point (P_{i_{1}}, …, P_{i_{l}}) ∈ V^{S}).

Definition 3.4. A subset W ⊂ V is called negligible if it is contained in a countable union of sets of the form p_{S}^{−1}(Z), where S ⊂ ℕ is a finite non-empty subset and Z ⊂ V^{S} is a Zariski closed subset such that Z ≠ V^{S}.

We will say that a property holds for a generic sequence of points of V if the set of sequences P ∈ V for which this property fails, is negligible.

Informally speaking, a property holds for a generic sequence P = { P_{1}, P_{2}, … } ∈ V if to ensure it it suffices to perform the following procedure countably many times: take a finite subsequence of hte sequence P and impose on it a Zariski open condition.

Of course, me must check whether negligible sets deserve their name. If the base field is countable, this is not the case, since any subset of V is negligible (V = ⋃_{x∈V} p_{{1}}^{−1}(x), and V is a countable set). The situation is better over uncountable fields:

Proposition 3.5. Any countable union of negligible sets is negligible, and V is not negligible provided that the base field k is uncountable.

Proof. Only the second assertion deserves proof. For the sake of brevity, let us give a proof only for a projective V over the complex numbers. Endow V = V^{ℕ} with the Tikhonov topology of direct product of classical topologies on V ’s. With this topology V becomes a compact Hausdorff space; each of p_{S}^{−1}(Z)’s is a closed nowhere dense subset of V, and the assertion follows from the Baire theorem.

Now we can prove the second part of Theorem 1.1.

Suppose that V ⊂ ℙ^{n} is an irreducible projective variety and that {P_{t}}_{t∈ℕ} is a generic sequence of distinct points of V; put X_{i} = { P_{1}, …, P_{i} } ⊂ V. We are to show that for any j ≥ 0 there exists an integer N > 0 such that b_{rs}(X_{i}) = b_{rs}(V) whenever s ≤ j and i ≥ N.

Proof. It suffices to prove that, for a fixed j > 0 and a generic sequence P ∈ V, we have

lim_{n→∞} dim ⟨v_{j}(X_{n})⟩ = dim ⟨v_{j}(X)⟩ .

It is clear that, for any projective variety X ⊂ ℙ^{N} and any l > dim ⟨X⟩, the set

{ (P_{1}, …, P_{l}) ∈ X^{l} | ⟨P_{1}, …, P_{l}⟩ ≠ ⟨X⟩ }

is contained in a Zariski closed proper subset of X^{l}, which yields our assertion.

Remark 3.6. Our definition of negligible subsets is admittedly coarse: we allow pretty large sets to be negligible. However, Theorem 1.1 is not likely to hold in higher dimensions if we replace Definition 3.4 by a more economical one. Let us, for example, say that a set W ⊂ V is “strongly negligible” if it is a countable union of subsets of the form ∏_{i∈ℕ} Z_{i}, where each Z_{i} is Zariski closed in V and Z_{i} ≠ V for almost all i ∈ N. Then the set of sequences P ∈ 𝕍 for which the assertion of Theorem 1.1 fails is not strongly negligible if dim V > 1. Indeed, suppose that the ground field is uncountable and (for the sake of brevity) that X is swept out by lines. Then for any sequence Z_{i} ⊂ V there exists a line L ⊂ V and an N ∈ ℕ such that L \ ⋃_{i≥N} Z_{i} is infinite. Hence, one may pick a sequence P = {P_{i}} such that P_{i} ∈ L \ W_{i} provided that i ≥ N. For this sequence P, graded Betti numbers of X_{i} will converge to graded Betti numbers of L rather than of X.

4. Lemmas on graded Betti numbers

In this section some auxiliary results on graded Betti numbers of (chiefly finite) subsets of ℙ^{n} are gathered. I believe that most of these facts are known to the experts and belong to the folklore.

We begin with the following fact.

Lemma 4.1. Suppose that X ⊂ ℙ^{n} is a closed subscheme, I_{X} is the ideal sheaf of X, and c ∈ ℤ. Then H^{t}(ℙ^{n}, I_{X}(c)) is isomorphic to H_{n+1−t}(K_{·}), where K_{i} = H^{n}(ℙ^{n}, { \atop L_{i}(X)}(c)) for i ∈ [1; n], all the other K_{j}’s are zero, and differentials K_{i} → K_{i−1} are induced by differentials in the resolution L_{·}(X).

The next proposition relates the homogeneous coordinate ring of a projectively normal variety to that of its hypersurface section.

Proposition 4.2. Suppose that X ⊂ ℙ^{n} is a projectively normal closed subscheme and V ⊂ ℙ^{n} is a hypersurface of degree d that does not meet Ass(X). Then R_{X∩V} ≅ R_{X}/(F), where F is the equation of V.

Proof. The set Ass(R_{X}) consists of homogeneous graded ideals corresponding to the points of Ass(X), plus possibly the maximal homogeneous ideal 𝔪 = (T_{0}, …, T_{n}). Since the homogeneous ideal I(X) ⊂ R is saturated, we see that 𝔪 ∉ Ass(R_{X}), whence F is not a zerodivisor in R_{X}. Hence, depth(R_{X}/(F)) = depth(R_{X}) − 1; since X is projectively normal, the right-hand side of this equality is positive, whence the ideal I_{X} + (F) ⊂ R is saturated and coincides with the homogeneous ideal of the subscheme X ∩ V. The proposition is proved.

Corollary 4.3. Under the assumptions of the above proposition, the minimal resolution L_{·}(X∩V) is isomorphic to the cone of the morphism

L_{·}(X)(−d) ⟶^{F} L_{·}(X), and b_{ij}(X ∩ V) = b_{ij}(X) + b_{i−1,j−d+1}(X), where

d = deg F.

Proof. Both assertions follow immediately from the exact sequence

0 → R_{X}(−d) ⟶^{F} R_{X} → R_{X∩V} → 0,

which exists by virtue of Proposition 4.3 .

In the sequel we will need to know graded Betti numbers of degenerate subvarieties of ℙ^{n}. These numbers are easily computable with the aid of Proposition 4.2. Subsequently we will use the following notation: if X ⊂ ℙ^{n−1} ⊂ ℙ^{n} is a closed subset, then b_{ij}(X, ℙ^{n−1}) (resp. b_{ij}(X, ℙ^{n})) denotes a graded Betti number of X regarded as a closed subset of ℙ^{n−1} (resp. ℙ^{n}).

Proposition 4.4. If X ⊂ ℙ^{n−1} is a closed subset and ℙ^{n−1} is a hyperplane in ℙ^{n}, then b_{ij}(X, ℙ^{n}) = b_{ij}(X, ℙ^{n−1}) + b_{i−1,j}(X, ℙ^{n−1}).

Proof. We may and will assume that ℙ^{n−1} ⊂ ℙ^{n} is defined by the equation T_{n} = 0; let R′_{X} denote the homogeneous coordinate ring of R as subvariety of ℙ^{n−1}. If p ∈ ℙ^{n} is a point with homogeneous coordinates (0 : ⋯: 1) and X̄ ⊂ ℙ^{n} is the cone over X with the vertex p, then RX̄ ≅ R′_{X} ⊗_{k}k[T_{n}]. Hence, the minimal resolution for the R-module RX̄ may be obtained from the minimal resolution for the k[T_{0}, …, T_{n−1}]- module R′_{X} by tensoring with k[T_{n}]. In particular, b_{ij}(X̄) = b_{ij}(X) for all i, j. Since X is a hyperplane section of X̄, everything follows now from Proposition 4.2.

Remark 4.5. Since the minimal resolution for RX̄ over R has the same length as the minimal resolution for R′_{X} over k[T_{0}, …, T_{n−1}], this length is at most n − 1, whence X̄ is projectively normal. Thus, all cones are projectively normal. I do not know a geometric proof of this simple fact. The following argument, which is due to F.L. Zak, shows at least that X̄ is linearly normal. Suppose that π: Ỹ → X̄ is an isomorphic projection from ℙ^{n+1}. Since X̄ is swept out by lines passing through p and lines are linearly normal, we infer that Y is swept out by lines passing through p̃ = π^{−1}(p). Hence, the embedded Zariski tangent space to X̄ at p (resp. to Y at p̃) coincides with the linear span of X̄ (resp. Y). Since π induces an isomorphism of Zariski tangent spaces, we see that linear spans of X̄ and Y are of the same dimension, which proves linear normality.

Propositions 4.2 and 4.4 have an amusing consequence:

Proposition 4.6. Suppose that Y ⊂ ℙ^{n+1} is a projective variety and that X = Y ∩ ℙ^{n} is its hyperplane section. If either Y or X is projectively normal, then b_{ij}(X, ℙ^{n}) = b_{ij}(Y, ℙ^{n+1}) for all i and j.

Proof. Since any extension of a projectively normal variety is projectively normal, we may and will assume that Y is projectively normal. Hence, the following equalities hold for all i and j:

b_{ij}(X, ℙ^{n+1}) =b_{ij}(Y) + b_{i−1,j}(Y)  (Corollary 4.3);
b_{ij}(X, ℙ^{n+1}) =b_{ij}(X, ℙ^{n}) + b_{i−1,j}(X, ℙ^{n}) (Proposition 4.4).

Since b_{0j}(X, ℙ^{n}) = b_{0j}(Y, ℙ^{n+1}) for all j, the proposition follows from the above equalities by induction on j.

Proposition 4.7. Let Y ⊂ ℙ^{n} be a closed subset and p ∈ ℙ^{n} a point outside X. Suppose that a number r ∈ ℕ is such that v_{r−1}(p) ∈ ⟨v_{r−1}(Y)⟩ and v_{r}(p) ∉ ⟨v_{r}(Y)⟩. Then the natural homomorphisms H_{s}^{l}(Y) → H_{s}^{l}(Y ∪ p) are isomorphisms whenever s does not equal r or r − 1, and for each l ≥ 0 there is an exact sequence

0 → H_{r}^{l}(Y) → H_{r}^{l}(Y ∪ p) → H_{0}^{l}(p) → H_{r−1}^{l+1}(Y) → H_{r−1}^{l+1}(Y ∪ p) → 0. (4.1)

Proof. The construction we use is present (in a dualized form and without mentioning the geometric meaning of the number r) in [10, section 2].

It follows immediately from the hypothesis and Corollaries 2.10 and 2.11 that there exists an exact sequence of graded modules

0 → J(Y) → J(Y ∪ p) → J(p)(−r) → 0.

The long exact sequence of Ext’s yields exact sequences

⋯ → H_{j−r+1}^{l}(p) → H_{j}^{l}(Y) → H_{j}^{l}(Y ∪ p) →

H_{j−r}^{l}(p) → H_{j−1}^{l+1}(Y) → H_{j−1}^{l+1}(Y ∪ p) → ….

Since H_{j}^{m}(p) = 0 for all j ≠ 0, the proposition follows.

Corollary 4.8. If a closed set Y ⊂ ℙ^{n−1} ⊂ ℙ^{n} spans the hyperplane ℙ^{n} and if p ∈ ℙ^{n} \ℙ^{n−1}, then b_{m1}(Y ∪p) = b_{m1}(Y, ℙ^{n−1})+b_{m−1,1}(Y, ℙ^{n−1})+ \binom{n}{m}, and b_{mj}(Y ∪ p) = b_{mj}(Y, ℙ^{n}) if j ≠ 1.

Proof. We may apply Proposition 4.7 with r = 1. Since b_{i0}(Y ∪ p) = 0 for i > 0 by virtue of Corollary 3.3, the exact sequence (4.1) with j = 1 boils down to

0 → H_{1}^{i}(Y) → H_{1}^{i}(Y ∪ p) → H_{0}^{i}(p) → 0.

Applying Proposition 4.4 and recalling that b_{i0}(p, ℙ^{n}) = \binom{n}{i}, we obtain the desired formula.

Corollary 4.9. Under the assumptions of Proposition 4.7, b_{nr}(Y ∪ p) = b_{nr}(Y) + 1.

Proof. Put l = n in the exact sequence (4.1) and recall that b_{n0}(p) =

I have not found a reference for the next proposition, but I believe that it is known to the experts.

Proposition 4.10. If X_{n} ⊂ ℙ^{n−1} is a set of n points spanning ℙ^{n−1}, then its graded Betti numbers are as follows: b_{00}(X_{n}) = 0, b_{m1}(X_{n}) = m\binom{n}{m+1} for m ∈ [1; n − 1], and all the other b_{ij}’s are zero.

Proof. We proceed by induction on n. For n = 2 the proposition is obvious. To make the induction step observe that X_{n+1} ⊂ ℙ^{n} is the union of X_{n} ⊂ ℙ^{n−1} ⊂ ℙ^{n} and a point outside ℙ^{n−1}. Applying Corollary 4.8 and Proposition 4.4, we obtain the equalities

b_{m1}(X_{n+1}) = b_{m1}(X_{n}) + b_{m−1,1}(X_{n}) + \binom{n}{m} =

m (\binom{n}{m + 1} + \binom{n}{m}) = m\binom{n + 1}{m + 1}.

This completes the induction and the proof.

Corollary 4.11. If X ⊂ ℙ^{N} is a non-degenerate variety of degree n and dimension N − n + 1, then b_{00}(X) = 1, b_{m1}(X) = m\binom{n}{m+1} for m ∈ [1; n − 1], and all the other b_{ij}’s are zero.

Proof. If X is a rational normal curve, then all follows from Proposition 4.6 since rational normal curves are projectively normal. If dim X > 1, then X may be obtained from a rational normal curve by a sequence of extensions, and the result follows by induction from Proposition 4.6 and the case dim X = 1.

The next proposition will be used in section 6. For the case of finite X it is essentially contained in [7, Lemma 1.2].

Proposition 4.12. For any closed subscheme X ⊂ ℙ^{n} and any i ∈ [1; n], put u_{i}(X) = max{ j | b_{ij}(X) ≠ 0 }. If H^{l}(I_{X}(t)) = 0 for all l > 1 and all t ≥ −n, then the sequence u_{1}, u_{2}, … u_{n} is non-decreasing.

Proof. Put v_{0} = 0 and v_{j} = u_{j} + j for any j ∈ [1; n]. To prove the lemma, we are to show that v_{0} < v_{1} ⋯ < v_{n}. Assume the converse. Then there exists a number s ∈ [1; n − 1] such that v_{s} ≥ v_{s+1} and v_{s} ≥ v_{s−1}. Put c = v_{s} − n − 1 in Lemma 4.1. Since v_{j} equals the maximal number m for which R(−m) is present (as a direct summand) in L_{j}(X), K_{j} ≠ 0 and that the differentials ∂_{j+1} : K_{j+1} → K_{j} and ∂_{j}: K_{j} → K_{j−1} are zero (we use the notation from Lemma 4.1). Indeed, if R(−v_{s}) is not present in L_{s+1} or L_{s−1}, then the vanishing of ∂_{j+1} and ∂_{j} is obvious, and if it is present, we observe that minimality of the resolution implies that the induced homomorphisms R(−v) → R(−v), where the two R(−v)’s are direct summands of adjacent terms of the resolution, should be zero.

Hence, H^{n+1−s}(I_{X}((v_{s} − n − 1)) = H_{s}(K_{·}) ≠ 0. Since v_{s} ≥ 1 and s ≤ n − 1, this contradicts hypothesis of the proposition.

5. Some evidence for the periodicity conjecture

Suppose that C ⊂ ℙ^{n} is a curve and that P = {P_{m} ∈ C}_{m∈ℕ} is a generic sequence of points. Proposition 5.1 below shows that at least lower right corners of Betti diagrams for the sets X_{m} = { P_{1}, …, P_{m} } behave as predicted by Conjecture 1.2.

Proposition 5.1. Suppose that C ⊂ ℙ^{n} is an irreducible curve of degree d and that {P_{i}}_{i∈ℕ} is a generic sequence of distinct points of C; put X_{i} = { P_{1}, …, P_{i} } ⊂ C. For any i, put a_{i} = max{ l | b_{nl}(X_{i}) ≠ 0 }. Then there exists a number N such that for any i ≥ N we have:

\begin{aligned}a_{i+d} & = a_{i} + 1; \\ b_{n,a_{i}}(X_{i}) & = b_{n,a_{i}+1}(X_{i+d}). \end{aligned} (5.1)

Remark 5.2. Proposition 4.12 shows that b_{n,a_{i}}(X_{i}) indeed stands in the lower right corner of the Betti diagram of X.

Proof. We start with some notation.

Notation 5.3. If X ⊂ ℙ^{n} is a closed subset and I_{X} ⊂ O_{ℙ^{n}} is its sheaf of ideals, put β(X) = max { t ∈ ℤ | H^{1}(ℙ^{n}, I_{X}(t)) ≠ 0 }.

Lemma 5.4. Suppose that X ⊂ ℙ^{n} is a closed subset. Then:

(a) If b_{nj}(X) ≠ 0, then H^{1}(ℙ^{n}, I_{X}(j − 1)) ≠ 0.

(b) If X is not projectively normal, then β(X) = l − 1, where l = max { t ∈ ℤ | b_{nt}(X) ≠ 0 }; moreover, h^{1}(ℙ^{n}, I_{X}(β(X))) = b_{nl}.

Proof of the lemma. We start with part (a). Lemma 4.1 shows that

H^{1}(I_{X}(t)) ≅ Ker(H^{n}(L̃_{n}(t)) → H^{n}(L̃_{n−1}(t))). (5.2)

Observe that the module L_{n} contains R(−n − j) as a direct summand and that, if ∂: L_{n} → L_{n−1} is the homomorphism from the resolution, then minimality of the resolution implies that ∂(R(−n−j)) is contained in a submodule of L_{n−1} that is a sum of R(−m_{i})’s with m_{i} > −n − j. Hence, the induced homomorphism H^{n}(L̃_{n}(s)) → H^{n}(L̃_{n−1}(s)) is zero when restricted to H^{n}(R̃(−n − 1)). This provides a non-zero element in the kernel in the right-hand side of (5.2) and proves part (a) of the proposition.

To prove part (b), observe that it follows from (a) that β(X) ≥ l −1; on the other hand, if t ≥ l, then L̃_{n}(t) is a direct sum of O(m_{i})’s with m_{i} ≥ −n, and it follows from (5.2) that h^{1}(I_{X}(t)) = 0. To compute h^{1}(I_{X}(l − 1)), observe that L̃_{n}(l − 1) equals (O(−n − 1))^{b_{nl}} plus a sum of O(m_{i})’s with m_{i} ≥ −n. The same argument as in the proof of the assertion (a) shows that ∂ maps H^{n}((O(−n − 1)))^{b_{nl}} to zero, which finishes the proof.

We proceed with the proof of the proposition. If X ⊂ ℙ^{n} is a finite subset, then β(X) equals the maximal t for which the set v_{t}(X) does not consist of linearly independent points, and h^{1}(I_{X}(β(X))) = card(X) − 1 − dim ⟨v_{β(X)}(X)⟩.

Lemma 5.5. If V ⊂ ℙ^{n} is a projective variety, then for a generic sequence P = {P_{i}}_{j∈ℕ} of distinct points of V one has

(5.3) dim ⟨X_{m}⟩ = min(m − 1, dim ⟨V ⟩)

for all m, where X_{m} = { P_{1}, …, P_{m}}.

Proof of the lemma. For any fixed m, the set

Z = { (P_{1}, …, P_{m}) ∈ V^{m} | equality (5.3) does not hold }

is closed in V^{m}, and Z ≠ V^{m}. Hence, to ensure that (5.3) holds for all m it suffices to impose a countable number of algebraic inequalities on finite subsequences of the sequence P. This is just Definition 3.4.

Applying the above lemma to all Veronese images of the curve C we see that, for a generic sequence P and for any l, m ∈ ℕ,

dim ⟨v_{l}(X_{m})⟩ = min(m − 1, dim ⟨v_{m}(C)⟩).

Choose a number l ∈ ℕ such that, for all t ≥ l, the equality dim ⟨v_{t}(C)⟩ = td − g holds, where g denotes arithmetical genus of C. I claim that equalities (5.1) hold for a generic P and for all i > ld + 1. Indeed, dim ⟨v_{m}(C)⟩ ≤ md for all m. Hence, for a generic P and for i > ld + 1, the set v_{m}(X_{i}) consists of linearly dependent points whenever m < l, and β(X_{i}) ≥ l. Thus, β(X_{i}) = ⌊(g + i − 1)/d⌋, and h^{1}(I_{X_{i}}(β(X_{i}))) = i − 1 + g − β(X_{i}) · d. This proves (5.1) and the proposition.

Remark 5.6. The above proof, which used linearity of Hilbert functions of curves, indicates that Conjecture 1.2 should not be true for higherdimensional varieties.

Another evidence for Conjecture 1.2 is provided by normal rational curves. The following proposition shows that for these curves the conjecture is true for arbitrary sequences of distinct points.

Proposition 5.7. Suppose that C ⊂ ℙ^{n} is a rational normal curve of degree n and that p = {P_{m}}_{m∈ℕ} is an arbitrary sequence of distinct points of C. Put X_{m} = { P_{1}, …, P_{m} }. Then b_{ij}(X_{m}) = b_{i,j+1}(X_{m+n}) whenever j ≥ 2.

Proof. We use the notation introduced in the statement of Conjecture 1.2. Let I_{l} ⊂ R_{C} stand for the homogeneous ideal of the closed subset X_{l} ⊂ C. If H is the hyperplane spanned by P_{m+1}, …, P_{m+n} and L ∈ (R_{C})_{1} is the image of its equation in R_{C}, then I_{m+n} = I_{m} ∩ (L). Since the points are distinct, L is not a zerodivisor in R/I_{m}. Hence, multiplication by L induces an isomorphism R_{C}/I_{m} ≅ (L)/I_{m} ∩ L. Taking grading into account, we obtain an exact sequence

0 → (R_{C}/I_{m})(−1) ⟶^{L} R_{C}/I_{m+n} → R_{C}/(L) → 0,

which gives rise to the long exact sequence

(5.4) ⋯ → Tor_{i}^{R}(k, R_{X_{m}})(−1) → Tor_{i}^{R}(k, R_{X_{m+n}}) →

Tor_{i}^{R}(k, R_{C}/(L)) → Tor_{i−1}^{R}(k, R_{X_{m}})(−1) → ….

Put Y = { P_{m+1}, …, P_{m+n} }. Since C is projectively normal, Proposition 4.2 says that R_{C}/(L) ≅ R_{Y}. Hence, (5.4) yields the following exact sequences:

⋯ → [Tor_{i}^{R}(k, R_{X_{m}})]_{i+j−1} → [Tor_{i}^{R}(k, R_{X_{m+n}})]_{i+j} →

[Tor_{i}^{R}(k, R_{C}/(L))]_{i+j} → [Tor_{i−1}^{R}(k, R_{X_{m}})]_{i+j−1} → ….

Propositions 4.10 and 4.4 show that [Tor_{a}^{R}(k, R_{Y})]_{a+b} = 0 for b ≠ 0, 1. Hence, b_{ij}(X_{m}) = b_{i,j+1}(X_{m+n}) for j ≥ 2. This proves our proposition.

6. Some explicit computations

In this section we compute graded Betti numbers of some finite subsets X ⊂ C, where C is a rational normal curve of degree n in ℙ^{n} and card(X) ≡ 0, 1 or 2 (mod n).

The case card(X) ≡ 0 (mod n) is the easiest:

Proposition 6.1. Suppose that C ⊂ ℙ^{n} is a rational normal curve of degree n and that X ⊂ C is a set of mn points, where m > 1. Then the graded Betti numbers of X are as follows: b_{i1}(X) = b_{i1}(C) = i\binom{n}{i+1} for any i ∈ [1; n − 1], b_{1,m−1}(X) = 1, b_{im}(X) = (i − 1)\binom{n}{i} for all i ∈ [2; n], b_{00}(X) = 1, and all the other b_{ij}’s are zero.

Proof. Decompose X into a disjoint union of m subsets containing n points each. If H_{i} is the linear span of the i-th subset, then X is a scheme-theoretic intersection of C with the hypersurface H_{1} ∪⋯∪H_{m}. Now everything follows from Corollary 4.3 and Corollary 4.11.

In the sequel P will always denote Hilbert polynomial of ℙ^{n}, i. e. P (t) = \binom{n+t}{t}.

Proposition 6.2. Let m > 1 be an integer and C ⊂ ℙ^{n} be a rational curve of degree n in ℙ^{n}. Suppose that X ⊂ C is a set of mn + 1 points. Then the graded Betti numbers of X are as follows: b_{i1}(X) = b_{i1}(C) = i\binom{n}{i+1} for any i ∈ [1; n − 1], b_{sm}(X) = n · \binom{n−1}{s−1} for any s ∈ [1; n], and all the other b_{ij}’s with i > 0 are zero.

Proof. Since v_{m}(X) is a set of mn + 1 linearly independent points that span ℙ^{mn} = ⟨v_{m}(C)⟩, Proposition 3.2 shows that b_{ij}(X) = b_{ij}(C) whenever j < m, and Lemma 5.4 shows that b_{nj}(X) = 0 for j > m. Since any finite subset of ℙ^{n} satisfies the hypotheses of Proposition 4.12, this proposition shows that b_{ij}(X) = 0 whenever j > m.

It remains to calculate b_{sm}’s for s < n. To that end observe that, denoting Hilbert polynomial of X ⊂ ℙ^{n} by P_{X}(·), we have the following formula:

P_{X}(t) − P_{C}(t) = ∑_{s=1}^{n}(−1)^{s}b_{sm}P (t − m − s). (6.1)

The left-hand side equals mn − nt; since deg P = n, the n polynomials P (· − m − s), where s ∈ [1; n], are linearly independent and the equation (6.1) has a unique solution in b_{1m}, …, b_{nm}. Thus, to finish the proof it suffices to verify that equation (6.1) holds for b_{sm} = \binom{n−1}{s−1}. Dividing both sides of (6.1) by n, we see that this is equivalent to saying that

∑_{s=1}^{n}(−1)^{s}\binom{n − 1}{s − 1}P (t − m − s) = m − t;

changing signs in both sides and substituting t = m + v + 1, we see that the above equality is equivalent to

∑_{j=0}^{n−1}(−1)^{j}\binom{n − 1}{j}P (v − j) = v + 1. (6.2)

The right-hand side of (6.2) coincides with the Hilbert polynomial P_{ℙ^{1}}(v); if we calculate the same Hilbert polynomial via the Koszul resolution that corresponds to the representation of ℙ^{1} as a complete intersection of n − 1 hyperplanes in ℙ^{n}, we arrive at the left-hand side of (6.2). This completes the proof.

Remark 6.3. Putting this proposition together with Proposition 4.10, we see that we know graded Betti numbers of all finite subsets X ⊂ C with card(X) ≡ 1 (mod n).

Proposition 6.4. Suppose that C ⊂ ℙ^{n} is a normal rational curve of degree n and X ⊂ C is a finite set consisting of mn + 2 points, where m > 1. Then b_{i1}(X) = b_{i1}(C) = i\binom{n}{i+1} for i ∈ [1; n − 1], b_{im}(X) = b_{n−i,1}(X) for all i ∈ [1; n − 1], b_{n,m+1}(X) = b_{00}(X) = 1, and all the other graded Betti numbers of X are zero.

Proof. Fix a point p ∈ X and put Y = X \p. Since ⟨v_{m}(X)⟩ = ⟨v_{m}(Y)⟩ and ⟨v_{m+1}(X)⟩ ≠ ⟨v_{m+1}(Y)⟩, Proposition 4.7 applies with r = m + 1. Taking into account Proposition 6.2 we see that b_{lj}(X) = b_{lj}(C) for j < m and that b_{lj}(X) = 0 for j > m + 1; Corollary 4.9 shows that b_{n,m+1}(X) = 1; hence, only b_{jm}(X) for j ∈ [1; n] and b_{j,m+1}(X) for j ∈ [1; n − 1] remain unknown.

To find them, put

a_{j} = b_{jm}(X) + b_{j−1,m+1}(X).

If P (t) denotes Hilbert polynomial of ℙ^{n}, i. e. \binom{n+t}{n}, then Hilbert polynomial of X equals

P_{C}(t) + ∑_{i=1}^{n}(−1)^{j}a_{j}P (t − j − m), (6.3)

where P_{C} denotes Hilbert polynomial of C. Since a_{n+1} = b_{n,m+1}(X) = 1, we have an identity

∑_{i=1}^{n}(−1)^{i}a_{j}P (· − j − m) = (−1)^{n}P (· − n − m − 1) + mn + 2 − P_{C}(·). (6.4)

Since deg P = n, the n polynomials P (· − j − m), where j ∈ [1; n], are linearly independent. Hence, equation (6.4) has a unique solution in a_{1}, …, a_{n} (if any). I claim that the numbers a_{j} = (n − j)\binom{n}{n−j+1} = b_{n−j,1}(C) provide such a solution. Indeed, substituting these numbers in (6.3) one obtains

\underbrace{P (t) + ∑_{i=1}^{n−1}(−1)^{i}b_{i1}(C)P (t − i − 1)}_{P_{C}(t)}

+\underbrace{(−1)^{n}(P (t − n − 1 − m) + ∑_{i=1}^{n−1} b_{i1}(C)P (t − n − 1 − (m − 1 − i))).}_{Q(t)}

Since P satisfies the identity (−1)^{n}P (−n − 1 − u) = P (u), we see that Q(t) = P_{C}(m − t) and P_{C}(t) + Q(t) = nt − 1 + n(m − t) + 1 = mn + 2, which coincides with Hilbert polynomial of X. Since equation (6.4) is equivalent to the statement that (6.3) equals Hilbert polynomial of X, our claim is proved. In particular, we proved that a_{n} = 0 = b_{nm}(X) + b_{n−1,m+1}(X). Hence, both of the summands are zero. Now Proposition 4.12 implies that b_{j,m+1}(X) = 0 for all j < n. Hence, b_{jm}(X) = a_{j} for all j, which completes our computation.

Remark 6.5. The previous proposition does not account for the case card(X) = n + 2. The same method applies in this case too, yielding the formulas b_{i1}(X) = b_{i1}(C) + b_{n−i,1}(C) for all i ∈ [1; n − 1], b_{00}(X) = b_{n2}(X) = 1, with all the other b_{ij}’s being zero. These formulas are contained in [9] and [8]. Applying Proposition 4.6, one obtains another proof of Theorem 1 from [8].

Appendix

In this appendix we reproduce a fragment from a log of a Macaulay session. The program computed (over the field of 31991 elements) minimal resolutions of finite subsets X_{m} ⊂ C, where C is a monomial curve C ⊂ ℙ^{3} given parametrically by t ↦ (1 : t : t^{4} : t^{5}) and X_{m} = { P_{1}, …, P_{m} }. The points P_{1}, … P_{32} correspond to the following values of the parameter t: 1, 74, 47, 20, 15, 18, 44, 93, 64, 39, 52, 70, 76, 77, 92, 46, 11, 41, 68, 62, 84, 5, 22, 32, 2, −6, 419, 606, 891, 300, 119, 684. In the tables that follow, betti resm means “Betti table of X_{m}”. Periodicity begins with m = 21.

1% betti res18
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     7    10     3
4:      -     -     2     2
1% betti res19
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     6     8     2
4:      -     1     4     3
1% betti res20
total:      1     8    13     6
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     5     6     2
4:      -     2     7     4
1% betti res21
total:      1    10    16     7
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     5    10     5
1% betti res22
total:      1     9    13     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -

3:      -     4     6     2
4:      -     4     7     2
5:      -     -     -     1
1% betti res23
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     3     4     1
5:      -     -     2     2
1% betti res24
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     2     2     -
5:      -     1     4     3
1% betti res25
total:      1     8    13     6
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     1     -     -
5:      -     2     7     4
1% betti res26
total:      1    10    16     7
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     -     -     -
5:      -     5    10     5
1% betti res27
total:      1     9    13     5
--------------------------------
0:      1     -     -     -

1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     -     -     -
5:      -     4     7     2
6:      -     -     -     1
1% betti res28
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     -     -     -
5:      -     3     4     1
6:      -     -     2     2
1% betti res29
total:      1     8    12     5
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     -     -     -
5:      -     2     2     -
6:      -     1     4     3
1% betti res30
total:      1     8    13     6
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2
4:      -     -     -     -
5:      -     1     -     -
6:      -     2     7     4
1% betti res31
total:      1    10    16     7
--------------------------------
0:      1     -     -     -
1:      -     1     -     -
2:      -     -     -     -
3:      -     4     6     2

 4:      -     -     -     -
 5:      -     -     -     -
 6:      -     5    10     5
1% betti res32
total:      1     9    13     5
--------------------------------
 0:      1     -     -     -
 1:      -     1     -     -
 2:      -     -     -     -
 3:      -     4     6     2
 4:      -     -     -     -
 5:      -     -     -     -
 6:      -     4     7     2
 7:      -     -     -     1

References

[1] E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris. Geometry of algebraic curves, volume 267 of Grundlehren der math. Wissenschaften. Springer, Berlin Heidelberg New York, 1984.

[2] D. Bayer and M. Stillman. Macaulay, a computer algebra program. http://www.math.columbia.edu/˜ bayer/Macaulay.html.

[3] C. Birkenhake. Noncomplete linear systems on abelian varieties. Trans. Amer. Math. Soc., 348(5):1885–1908, 1996.

[4] N. Bourbaki. Algèbre. Chapitre X. Algèbre homologique. Éléments de mathématique. Masson, Paris, 1980.

[5] D. Eisenbud. Green’s conjecture: an orientation for algebraists. In D. Eisenbud and C. Huneke, editors, Free Resolutions in Commutative Algebra and Algebraic Geometry, number 2 in Research Notes in Mathematics, pages 51–78. Jones and Bartlett Publishers, 1992.

[6] S. Goto and K. Watanabe. On graded rings, I. J. Math. Soc. Japan, 30(2):179– 213, 1978.

[7] M. L. Green and R. Lazarsfeld. Some results on the syzygies of finite sets and algebraic curves. Compositio Math., 67:301–314, 1988.

[8] L. T. Hoa. On minimal free resolutions of projective varieties of degree = codimension + 2. J. Pure Appl. Algebra, 87:241–250, 1993.

[9] P. Maroscia, U. Nagel, and W. Vogel. On the minimal free resolutions of finite sets of points in ℙ^{n}. C. R. Math. Rep. Acad. Sci. Canada, 12(1):27–32, 1990.

[10] U. Nagel. On the minimal free resolutions of r + 3 points in projective r-space. J. Pure Appl. Algebra, 96(1):23–38, 1994.

E-mail address: serge@main.mccme.rssi.ru, lvovski@mpim-bonn.mpg.de