Extremal Zerodimensional Subschemes of IP^{2}

by Bettina and Martin Kreuzer

Abstract. Given a 0-dimensional subscheme 𝕏 of ℙ^{2}, we bound the number of points in the support of 𝕏 which have maximal degree in 𝕏. For reduced schemes 𝕏, this yields a lower bound for the colength of the conductor F of the homogeneous coordinate ring R of 𝕏 in its integral closure \overline{R}. This bound is attained by Castelnuovo sets for which we calculate ℓ(R/F) explicitly.

Using the canonical decomposition of 𝕏, we also show a sharp upper bound for ℓ(R/F). Applications include estimates for the singularity degree ℓ(\overline{R}/R) and the superabundance ℓ(\overline{R}/R) − ℓ(R/F) of 𝕏.

Keywords. Zerodimensional scheme, Hilbert function, conductor colength, singularity degree

Introduction

For a 0-dimensional subscheme 𝕏 ⊆ ℙ^{2}, various numerical invariants can be readily defined and have been studied classically. For instance, G. Castelnuovo examined the postulation of point sets arising by taking hyperplane sections of space curves (cf. [1]). Nowadays, one refers to the Hilbert function of 𝕏 instead and calls its first difference function the Castelnuovo function ΔH_{𝕏} of 𝕏 (cf. [2]). All possible Castelnuovo functions have been classified. Among all 0-dimensional schemes 𝕏 having a specified Castelnuovo function, Castelnuovo sets are obtained by lifting lex-segment monomial ideals and have simple geometrical realizations (cf. 1.3, 1.4).

Other invariants come into play, if one looks at the Cayley-Bacharach property of 𝕏. If P ∈ Supp 𝕏 and 𝕐 ⊆ 𝕏 is a generic subscheme of colength one differing from 𝕏 only at P, we call the least degree of a curve containing 𝕐 but not 𝕏 the degree of P in 𝕏. The degrees of the points of a Castelnuovo set can be computed purely from its Castelnuovo function (cf. 1.9, 1.10). In fact, Castelnuovo sets are extremal among all 0-dimensional schemes with a fixed Castelnuovo function in that they have the maximal number of points of bounded degree. This is shown in section 2 and generalizes the corresponding statement for reduced schemes (cf. [13]). Since, generically, one expects that all points have maximal degree, it follows that Castelnuovo sets are very “special” 0-dimensional schemes.

On the algebraic side, given a reduced 0-dimensional subscheme 𝕏 ⊆ ℙ^{2} with homogeneous coordinate ring R, one may look at the integral closure \overline{R} of R in its full ring of quotients, and at the conductor ideal 𝔉 ⊆ R of R in \overline{R}. The first description of 𝔉 in terms of the degrees of the points was given in [12]. The colength ℓ(R/𝔉) of 𝔉 is another invariant of 𝕏 which was studied classically, e.g. by D. Gorenstein in a similar situation (cf. [3]). Using the formula

ℓ(R/𝔉) = ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) − ∑_{i≥0} i · ΔH_{𝕏}(i)

(cf. 4.4.b), we can show that for a given Castelnuovo function, the conductor colength is minimalized by Castelnuovo sets. We go on to provide a sharp upper bound for ℓ(R/𝔉) as well (cf. 4.5.b).

To prove this bound, we need to introduce and study the canonical decomposition of a reduced 0-dimensional subscheme 𝕏 ⊆ ℙ^{2} in section 3. Starting from a decomposition of the Castelnuovo function ΔH_{𝕏} into layers ΔH_{i} of decreasing type, one gets a corresponding decomposition 𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)} such that 𝕏^{(i)} has Castelnuovo function ΔH_{i} (cf. 3.1, 3.2). We relate the degrees of the points of 𝕏^{(i)} to their degree in 𝕏 (cf. 3.5) and construct for each Castelnuovo function a reduced 0-dimensional scheme 𝕏 whose canonical components are Cayley-Bacharach schemes, i.e. are such that deg_{𝕏^{(i)}}(P) is maximal for all P ∈ Supp 𝕏^{(i)} (cf. 3.6).

Finally we consider the singularity degree ℓ(\overline{R}/R) and the superabundance δ_{𝕏} = ℓ(\overline{R}/R) − ℓ(R/𝔉) = ℓ(\overline{R}/𝔉) − 2ℓ(R/𝔉) of 𝕏. The first invariant is nothing but the usual singularity degree of the affine cone over 𝕏 at its vertex, while the second one takes its name from the inequality 2ℓ(R/𝔉) ≤ ℓ(\overline{R}/𝔉) which was first shown in [9] for the local situation. In our graded case, we are able to refine this into δ_{𝕏} ≥ δ′_{h} ≥ 0 with a quantity δ′_{h} depending only on ΔH_{𝕏} (cf. 4.8). Moreover, the equality δ_{𝕏} = δ′_{h} characterizes Cayley-Bacharach schemes, and the equality δ′_{h} = 0 characterizes the symmetry of ΔH_{𝕏}, so that altogether δ_{𝕏} = 0 characterizes arithmetically Gorenstein schemes, in generalization of Gorenstein’s original result (cf. [3], Theorem 6). The other extreme for the superabundance is again provided by Castelnuovo sets (cf. 4.8.a, 4.9). Applications include a characterization of Castelnuovo functions ΔH_{𝕏} which force 𝕏 to be a Cayley-Bacharach scheme (cf. 2.5) resp. which force every canonical component of 𝕏 to be a Cayley-Bacharach scheme (cf. 4.10).

Acknowledgements. Parts of this paper are based on the first author’s master thesis under the supervision of A.V. Geramita (Kingston, Canada) and her diplom thesis under E. Kunz (Regensburg, Germany). The authors are very grateful to both for helpful conversations.

1. Castelnuovo Schemes

In this paper we work over an algebraically closed field K of arbitrary characteristic. By ℙ^{2} we denote the projective plane over K. For a 0-dimensional subscheme 𝕏 ⊆ ℙ^{2}, we let I_{𝕏} ⊆ K[X_{0}, X_{1}, X_{2}] be its saturated homogeneous vanishing ideal, and R = K[X_{0}, X_{1}, X_{2}]/I_{𝕏} its homogeneous coordinate ring. It is well-known that R = ⊕_{n≥0}R_{n} is a standard graded K-algebra whose Hilbert function H_{𝕏} : ℤ ⟶ ℕ (n ↦ dim_{K} R_{n}) satisfies H_{𝕏}(n) = 0 for n < 0 and

1 = H_{𝕏}(0) < H_{𝕏}(1) < … < H_{𝕏}(σ_{𝕏}) < deg 𝕏 = H_{𝕏}(σ_{𝕏} + 1) = H_{𝕏}(σ_{𝕏} + 2) = …

for some number σ_{𝕏} ≥ −1. We shall also consider the number α_{𝕏} = min{n ∈ ℕ | (I_{𝕏})_{n} ≠ 0}. It is well-known that σ_{𝕏} ≥ α_{𝕏} − 2.

We may choose the coordinates {X_{0}, X_{1}, X_{2}} such that the line Z(X_{0}) does not contain any point of Supp 𝕏. Then x_{0} = X_{0} + I_{𝕏} ∈ R_{1} is not a zerodivisor of the 1-dimensional Cohen-Macaulay ring R. Thus \overline{R} = R/(x_{0}) is a finite K-algebra. Its Hilbert function H\overline{R} = ΔH_{𝕏} : ℤ ⟶ ℕ (n ↦ H_{𝕏}(n) − H_{𝕏}(n − 1)) is called the Castelnuovo function of 𝕏. The following properties of ΔH_{𝕏} follow easily from Macaulay’s characterization of Hilbert functions of standard graded algebras (cf. e.g. [2], 2.1).

1.1. Proposition. a) ΔH_{𝕏}(n) = 0 for n < 0, b) ΔH_{𝕏}(n) = n + 1 if and only if n ∈ {0, …, α_{𝕏} − 1},

c) ΔH_{𝕏}(n) ≥ ΔH_{𝕏}(n + 1) for α_{𝕏} − 1 ≤ n ≤ σ_{𝕏} + 1,

d) ΔH_{𝕏}(σ_{𝕏} + 1) > 0, and

e) ΔH_{𝕏}(n) = 0 for n > σ_{𝕏} + 1.

Since, as we shall recall below, those properties actually characterize all possible functions ΔH_{𝕏}, we introduce the following notion.

1.2. Definition. Let α_{h} ≥ 1, σ_{h} ≥ α_{h} − 2, and h : ℤ ⟶ ℕ (n ↦ h_{n}) be such that a) h_{n} = 0 for n < 0,

b) h_{n} = n + 1 if and only if n ∈ {0, …, α_{h} − 1},

c) h_{n} ≥ h_{n+1} for α_{h} − 1 ≤ n ≤ σ_{h} + 1,

d) h_{σ_{h}+1} > 0, and

e) h_{n} = 0 for n > σ_{h} + 1.

Then h is called a Castelnuovo function with invariants α_{h} and σ_{h}.

Given a function h like this, we introduce the following 0-dimensional subschemes of ℙ^{2} which will be our objects of study throughout this section.

1.3. Definiton. Let h : ℤ ⟶ ℕ be a Castelnuovo function with invariants α_{h} and σ_{h}, and let {s_{0}, …, s_{σ_{h}+1}} ⊆ K, {t_{0}, …, t_{α_{h}−1}} ⊆ K be sets of pairwise distinct elements. The reduced 0-dimensional subscheme

𝕏(h) := {(1 : s_{i} : t_{j}) ∈ ℙ^{2} | 0 ≤ i + j ≤ σ_{h} + 1, 0 ≤ j < h_{i+j}}

is called the Castelnuovo set for h with parameters s_{0}, …, s_{σ_{h}+1} and t_{0}, …, t_{α_{h}−1}.

1.4. Example. Let char (K) = 0 and h : ℤ ⟶ ℕ such that α_{h} = 3, σ_{h} = 3, h_{0} = 1, h_{1} = 2, h_{2} = h_{3} = 3, and h_{4} = 1. Then, choosing s_{i} = i and t_{j} = j for i = 0, …, 4 and j = 0, 1, 2, we obtain the following Castelnuovo set 𝕏(h):

0 1 2 3 4 0 1 2  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}• • • • • • • • • •^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}

Obviously h_{i} is the number of points chosen on the diagonal x + y = i, starting with the one on the x-axis.

The ubiquity of Castelnuovo sets comes from Hartshorne’s method for lifting lexsegment monomial ideals (cf. [8]). An elementary proof of the next theorem follows from [4], 2.2.

1.5. Theorem. Given a Castelnuovo function h with invariants α_{h} and σ_{h}, as well as pairwise distinct elements s_{0}, …, s_{σ_{h}+1} ∈ K and t_{0}, …, t_{α_{h}−1} ∈ K, the Castelnuovo set 𝕏(h) for h with those parameters has Castelnuovo function ΔH_{𝕏} = h.

The ideal I_{𝕏} is obtained by replacing X_{1}^{a} with ∏_{i=0}^{a−1}(X_{1} − t_{i}X_{0}) and X_{2}^{b} with ∏_{i=0}^{b}(X_{1} − s_{i}X_{0}) in the lex-segment ideal J_{𝕏} corresponding to h. Thus we have J_{𝕏} = I_{𝕏} + (X_{0})/(X_{0}) = (f_{0}, …, f_{α_{h}}) with f_{i} = X_{1}^{m(i)−i}X_{2}^{i} and m(i) = min{n ≥ α_{h} | h_{n} ≤ i} for i = 0, …, α_{h}.

1.6. Example. Let us label the points with integer coordinates in Example 1.4 as follows:

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The points “∘” at the convex corners of the shaded area correspond to the monomials X_{1}^{5}, X_{1}^{3}X_{2}, X_{1}^{2}X_{2}^{2}, X_{2}^{3}. Since m(0) = 5, m(1) = 4, m(2) = 4, m(3) = 3, those monomials are exactly the generators f_{0}, …, f_{3} of J_{𝕏}.

In fact, Castelnuovo sets of the form 𝕏(h) do not only have the correct Hilbert function, but they form very special sets of points with that Hilbert function. In the sequel we shall show that for a number of invariants, they attain extremal values.

1.7. Definition. a) Let 𝕏 ⊆ ℙ^{2} be a 0-dimensional subscheme. If 𝕐 ⊆ 𝕏 is a subscheme of colength one, i.e. with deg 𝕐 = deg 𝕏 − 1, then the vanishing ideal I_{𝕐} of 𝕐 contains I_{𝕏} and its image I_{𝕐/𝕏} := \overline{I}_{𝕐} ⊆ R in the coordinate ring R of 𝕏 satisfies

dim_{K}(I_{𝕐/𝕏}) = \{ \begin{matrix}1 & for n ≥ α_{𝕐/𝕏}, \\ 0 & otherwise \end{matrix}

for some α_{𝕐/𝕏} ≥ 0. This number α_{𝕐/𝕏} is called the degree of 𝕐 in 𝕏.

b) A nonzero element of (I_{𝕐/𝕏})_{α_{𝕐/𝕏}} is called a minimal separator of 𝕐 in 𝕏.

c) For P ∈ Supp 𝕏, the degree of P in 𝕏 is defined as

deg_{𝕏}(P) := max{α_{𝕐/𝕏} | 𝕐 ⊆ 𝕏 subscheme of colength one, O_{𝕐,P} ≠ O_{𝕏,P} }.

1.8. Remarks. Let 𝕏 ⊆ ℙ^{2} be a 0-dimensional subscheme and 𝕐 ⊆ 𝕏 a subscheme of colength one with minimal separator f_{𝕐} ∈ R.

a) The ring S := R/(x_{0} − 1) is the affine coordinate ring of 𝕏 in 𝔸^{2} = D_{+}(X_{0}). Under the canonical isomorphism S ≅ H^{0}(𝕏, O_{𝕏}) ≅ ∏_{P ∈Supp 𝕏} O_{𝕏,P}, the residue classes (i.e. the dehomogenization w.r.t. X_{0}) of minimal separators of 𝕐 correspond one to one to elements of the form (0, …, 0, s_{P} , 0, …, 0) where s_{P} ∈ 𝔖(O_{𝕏,P}) = {r ∈ O_{𝕏,P} | r ·𝔪_{𝕏,P} = 0} is a nonzero socle element. The ideal of 𝕐 in S is K ·(0, …, 0, s_{p}, 0, …, 0).

b) Let P ∈ Supp 𝕏. Using the map Φ : R −↠^{ϵ} S = R/(x_{0} − 1) ≅ ∏_{Q∈Supp 𝕏} O_{𝕏,Q}

−↠^{proj.} O_{𝕏,P} −↠^{can.} O_{𝕏,P} /𝔪_{𝕏,P} = K we can form the value r(P) = Φ(r) ∈ K of r at P for

each r ∈ R. If r ∈ R_{n}, then

r · f_{𝕐} = r(P) · x_{0}^{n} · f_{𝕐}.

This follows from the fact that ϵ|_{R_{n}} is injective (cf. [10], section 1).

c) We have σ_{𝕐} ≤ σ_{𝕏} and α_{𝕐/𝕏} ≤ σ_{𝕏} + 1. In particular, 0 ≤ deg_{𝕏}(P) ≤ σ_{𝕏} + 1 for every P ∈ Supp 𝕏.

For Castelnuovo sets, we can determine the degree of each point in a simple way.

1.9. Proposition. Let h be a Castelnuovo function with invariants α_{h} and σ_{h}, and let 𝕏(h) be the Castelnuovo set with parameters s_{0}, …, s_{σ_{h}+1} and t_{0}, …, t_{α_{h}−1}. Then the point P = (1 : s_{p} : t_{q}) ∈ Supp 𝕏(h) has degree

deg_{𝕏(h)}(P) = max{n ≥ α_{h} | h_{n} > q}.

In particular, this degree depends only on q, i.e. all points of 𝕏(h) on the line Z(X_{2} − t_{q}X_{0}) have the same degree.

Proof. In the first step, we show that the point P = (1 : s_{0} : t_{0}) has maximal degree deg_{𝕏(h)}(P) = σ_{h} + 1. Suppose deg_{𝕏(h)}(P) ≤ σ_{h}. Then there is a curve of degree σ_{h} containing 𝕏(h) \ {P }, but not P. This curve meets the line Z(X_{2} − t_{0}X_{0}) in the σ_{h} + 1 points (1 : s_{1} : t_{0}), …, (1 : s_{σ_{h}+1} : t_{0}). By Bézout, it therefore has that line as a component and contains P, a contradiction.

In the general case, consider P = (1 : s_{p} : t_{q}) ∈ Supp 𝕏(h) with 0 ≤ p ≤ σ_{h} + 1 and 0 ≤ q ≤ α_{h} − 1. If q > 0, we define q lines L_{0} = Z(X_{2} − t_{0}X_{0}), …, L_{q−1} = Z(X_{2} − t_{q−1}X_{0}), and if p > 0, we define p lines V_{0} = Z(X_{1}−s_{0}X_{0}), …, V_{p−1} = Z(X_{1}−s_{p−1}X_{0}). Also, we let m := max{n ≥ α_{h} | h_{n} > q}, and we start by showing deg_{𝕏(h)}(P) ≤ m. Clearly, we have m ≥ p + q.

In case m = p + q, the form ∏_{i=0}^{p−1}(X_{1} − s_{i}X_{0}) ∏_{j=0}^{q−1}(X_{2} − t_{j}X_{0}) defines a curve of degree m containing 𝕏(h) \ {P }, but not P. In case m > p + q, we consider the subscheme of 𝕏(h) consisting of those points of 𝕏(h) which are not contained in one of the lines L_{0}, …, L_{q−1}, V_{0}, …, V_{p−1}. This subscheme is contained in the set

D = {(1 : s_{i} : t_{j}) | p ≤ i ≤ m − q, q ≤ j ≤ m − p, i + j ≤ m}.

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 _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  L_{0} L_{1} L_{q−1} V_{0} V_{1}  V_{p−1} P •  • •^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} D

The set D is a Castelnuovo set itself, and by the first step we find a form F of degree m − p − q containing D \ {P }, but not P. Then F · ∏_{i=0}^{p−1}(X_{1} − s_{i}X_{0}) ∏_{j=0}^{q−1}(X_{2} − t_{j}X_{0}) defines a curve of degree m containing 𝕏(h) \ {P }, but not P.

Finally, we show deg_{𝕏(h)}(P) = m. Suppose there is a curve of degree m − 1 containing 𝕏(h) \ {P }. Then, for i ∈ {1, …, q − 1} at least m + 1 − i points of 𝕏(h) are contained in L_{i}. By Bézout and induction, the lines L_{0}, …, L_{q−1} are components of that curve. The residual curve has degree m−q−1 and contains 𝕏(h)\({P }∪⋃_{i=0}^{q−1}(L_{i}∩ 𝕏(h))). In particular, the residual curve contains m−q points on the line Z(X_{2} −t_{q}X_{0}), i.e. by Bézout the whole line. Thus our curve vanishes at P and deg_{𝕏(h)}(P) = m. □

1.10. Example. In the situation of Example 1.4, i.e. if s_{i} = i and t_{j} = j, one can find the degree of a point P ∈ 𝕏(h) graphically as follows:

0 1 2 3 4 0 1 2  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}• • • • • • • • • •P^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}

This means, we start by drawing the line Z(X_{2} − t_{2}) and follow it to the rightmost point of 𝕏(h). Through that point we draw the diagonal line of slope −1. Its point of intersection with the X_{1}-axis has coordinate deg_{𝕏(h)}(P). In the picture, P = (1 : 1 : 1) has degree deg_{𝕏(h)}(P) = 3. In general, the degree of P = (1 : p : q) is #(𝕏(h) ∩ Z(X_{2} − qX_{1})) + q − 1.

1.11. Corollary. The Castelnuovo set 𝕏(h) contains exactly \binom{σ_{h}+3}{2} − \binom{σ_{h}+3−h_{σ_{h}+1}}{2} points of maximal degree σ_{h} + 1.

Proof. This follows by counting the number of points of 𝕏(h)_{σ_{h}+1} = {(1 : s_{i} : t_{j}) | 0 ≤ j ≤ h_{σ_{h}+1} − 1, 0 ≤ i ≤ σ_{h} + 1 − j} which are by 1.9 precisely the points of 𝕏(h) of degree σ_{h} + 1. □

2. Schemes with many points of low degree

For a 0-dimensional subscheme 𝕏 ⊆ ℙ^{2} and n ≥ 0, we introduce additional notations:

𝕏_{n} := {P ∈ Supp 𝕏 | deg_{𝕏}(P) = n}
𝕏_{≤n} := {P ∈ Supp 𝕏 | deg_{𝕏}(P) ≤ n}

Among all reduced 0-dimensional subschemes 𝕏 ⊆ ℙ^{2} with a fixed Hilbert function, the Castelnuovo set has the maximal number of points of degree ≤ n (cf. [13], 2.3.4). In this section we prove a generalization of that result to the nonreduced case. But first we need to compare the degrees of two subschemes 𝕐, 𝕐′ ⊆ 𝕏 of colength one and their intersection ℤ = 𝕐 ∩ 𝕐′.

2.1. Proposition. Suppose 𝕐, 𝕐′ ⊆ 𝕏 are two subschemes of colength one such that O_{𝕐,P} ≠ O_{𝕏,P} and O_{𝕐′,Q} ≠ O_{𝕏,Q} for some P, Q ∈ Supp 𝕏 with P ≠ Q. Then ℤ := 𝕐∩𝕐′ satisfies

a) deg ℤ = deg 𝕏 − 2,

b) α_{𝕐′/𝕏} − 1 ≤ α_{ℤ/𝕐} ≤ α_{𝕐′/𝕏}, and
c) if α_{𝕐/𝕏} ≠ α_{𝕐′/𝕏} then α_{ℤ/𝕐} = α_{𝕐′/𝕏}.

Proof. “a)” This follows from 1.8.a and the assumption P ≠ Q.

“b)” Let f_{𝕐′} be a minimal separator of 𝕐′, and \overline{f}_{𝕐′} its image in the ring R_{𝕐} = K[X_{0}, X_{1}, X_{2}]/I_{𝕐}. Since 𝕐 ≠ 𝕐′, we have \overline{f}_{𝕐′} ∈ I_{ℤ/𝕐} \ \{0\}, and therefore α_{ℤ/𝕐} ≤ α_{𝕐′/𝕏}. In case α_{𝕐/𝕏} ≠ α_{𝕐′/𝕏}, the remaining claim α_{𝕐′/𝕏} − 1 ≤ α_{ℤ/𝕐} follows from c), so we shall assume now α_{𝕐/𝕏} = α_{𝕐′/𝕏}. Let f_{𝕐} be a minimal separator of 𝕐 in 𝕏, and g_{ℤ} ∈ R a representative of a minimal separator of ℤ in 𝕐. Letting d := α_{𝕐′/𝕏} − α_{ℤ/𝕐′}, our hypothesis implies x_{0}^{d}g_{ℤ} = λf_{𝕐} + μf_{𝕐′} for some λ, μ ∈ K. If λ = 0, then d = 0, because I_{𝕐′/𝕏} = (f_{𝕐′}) is a saturated ideal, and we are done. Hence we can assume λ ≠ 0. Since \overline{g}_{ℤ} ∈ R_{𝕐} is a minimal separator of ℤ in 𝕐, we get from 1.8.b that x_{i}·g_{ℤ} = x_{i}(Q)·x_{0}·g_{ℤ}+h_{i} for some homogeneous element h_{i} ∈ I_{𝕐/𝕏} of degree α_{ℤ/𝕐} + 1. Suppose now d > 1. Then (I_{𝕐/𝕏})_{α_{ℤ/𝕐}+1} = 0 implies h_{i} = 0. Next we calculate

x_{i}x_{0}^{d}g_{ℤ} = x_{0}^{d}x_{i}(Q)x_{0}g_{ℤ} = x_{0}(x_{i}(Q)λf_{𝕐} + x_{i}(Q)μf_{𝕐′})
and  x_{i}x_{0}^{d}g_{ℤ} = λx_{i}f_{𝕐} + μx_{i}f_{𝕐′} = x_{0}(x_{i}(P)λf_{𝕐} + x_{i}(Q)μf_{𝕐′}),

hence x_{i}(P) = x_{i}(Q) for i = 1, 2. Thus P = Q, a contradiction.

“c)” Suppose d := α_{𝕐′/𝕏} − α_{ℤ/𝕐} > 0. In case δ := α_{𝕐′/𝕏} − α_{𝕐/𝕏} > 0 we get x_{0}^{d}g_{ℤ} = λx_{0}^{δ}f_{𝕐} + μf_{𝕐′} for some λ, μ ∈ K. Since x_{0} ∤ f_{𝕐′}, this implies μ = 0, and then g_{ℤ} ∈ I_{𝕐/𝕏} contradicts \overline{g}_{ℤ} ∈ R_{𝕐} \ \{0\}. In case δ < 0, we get x_{0}^{d}g_{ℤ} = μf_{𝕐′} for some μ ∈ K \ {0}, contradicting x_{0} ∤ f_{𝕐′}. Thus d = 0, as was to be shown. □

Now we are ready to prove the main result of this section.

2.2. Proposition. Let h be a Castelnuovo function, 𝕏(h) a Castelnuovo set for h, and 𝕏 ⊆ ℙ^{2} a 0-dimensional subscheme with ΔH_{𝕏} = h. Then

#𝕏_{≤n} ≤ #𝕏(h)_{≤n}  for all n ≥ 0.

Proof. We proceed by induction on s = deg 𝕏. In case s = 1, the scheme 𝕏 = {P } is a reduced point and deg_{𝕏}(P) = 0. In case s > 1 we suppose there is a subscheme 𝕏 ⊆ ℙ^{2} such that ΔH_{𝕏} = h and a minimal number d ≥ 0 such that #𝕏_{≤d} > #𝕏(h)_{≤d}. Then #𝕏(h)_{≤d} + 1 ≤ #𝕏_{≤d} and #𝕏_{≤d−1} ≤ #𝕏(h)_{≤d−1} imply #𝕏(h)_{d} + 1 ≤ #𝕏_{d}, i.e. there is a point P ∈ Supp 𝕏 such that deg_{𝕏}(P) = d. A nonzero socle element of O_{𝕏,P} defines a subscheme 𝕐 ⊆ 𝕏 of degree s − 1 with O_{𝕐,P} ≠ O_{𝕏,P} and with α_{𝕐/𝕏} = d. Thus the Castelnuovo fuction h′ of 𝕐 is given by

h′_{n} = ΔH_{𝕐}(n) = \{ \begin{matrix}h_{n} & if n ≠ d, \\ h_{n} − 1 & if n = d. \end{matrix}

By induction, #𝕐_{≤n} ≤ #𝕏(h′)_{≤n} for n ≥ 0. Therefore we get

#𝕏(h′)_{≤d} ≥ #𝕐_{≤d} ≥ #{Q ∈ Supp 𝕐 | Q ≠ P, deg_{𝕐}(Q) ≤ d}.

Consider a point Q ∈ Supp 𝕏 \ {P } and a subscheme 𝕐′ ⊆ 𝕏 of degree s − 1 such that O_{𝕐′,Q} ≠ O_{𝕏,Q} and α_{ℤ/𝕐} = deg_{𝕐}(Q) for ℤ = 𝕐 ∩ 𝕐′. Then, by 2.1.b, we have deg_{𝕐}(Q) = α_{ℤ/𝕐} ≤ α_{𝕐′/𝕏} ≤ deg_{𝕏}(Q), and we conclude

\begin{aligned}#𝕏(h′)_{≤d} & ≥ #\{Q ∈ Supp 𝕏 | Q ≠ P, deg_{𝕏}(Q) ≤ d\} = #𝕏_{≤d} − 1 \\ & > #𝕏(h)_{≤d} − 1. \end{aligned}

From the above description of h′ we get 𝕏(h′) = 𝕏(h) \ {P′}, where P′ = (1 : s_{d−h_{d}+1} : t_{h_{d}−1}). Since h′ is a Castelnuovo function again, we also get h_{d+1} = h′_{d+1} ≤ h′_{d} = h_{d} −1, so that we can calculate the degree of P′ in 𝕏(h) using 1.9. We find

deg_{𝕏(h)}(P′) = max{n ≥ α_{h} | h_{n} > h_{d} − 1} = d.

For every point Q ∈ Supp 𝕏(h)\{P′} with deg_{𝕏(h)}(Q) ≠ d, Prop. 2.1.c yields deg_{𝕏(h)}(Q) = deg_{𝕏(h′)}(Q). For points Q ∈ Supp 𝕏(h) \ {P′} with deg_{𝕏(h)}(Q) = d, Prop. 2.1.b yields deg_{𝕏(h′)}(Q) ≤ d. Altogether, #𝕏(h′)_{≤d} = #𝕏(h)_{≤d} − 1, contradicting the above inequality. □

Note that obviously the inequalities #𝕏_{n} ≤ #𝕏(h)_{n} cannot hold for all n ≥ 0, because ∑_{n≥0} #𝕏_{n} = deg 𝕏 = ∑_{n≥0} #𝕏(h)_{n} for reduced schemes 𝕏. When dealing with nonreduced subschemes 𝕏 ⊆ ℙ^{2}, we could hope for the stronger inequality

∑_{P ∈Supp 𝕏,deg_{𝕏}(P)≤n} dim_{K} O_{𝕏,P} ≤ #𝕏(h)_{≤n}

in which each point P ∈ Supp 𝕏 of degree ≤ n is weighted by its multiplicity dim_{K} O_{𝕏,P}. However, the following example shows that this stronger inequality does not hold in general.

2.3. Example. Let 𝕏 = Z(X_{1}^{2}, X_{1}X_{2}, X_{2}^{3} − X_{0}^{2}X_{1}) ⊆ ℙ^{2}. Then Supp 𝕏 = {P } with P = (1 : 0 : 0) and deg 𝕏 = 4. The socle of O_{𝕏,P} = K[X_{1}, X_{2}]/(X_{1}^{2}, X_{1}X_{2}, X_{2}^{3} − X_{1}) is K · \overline{X}_{1}, so the only subscheme 𝕐 ⊆ 𝕏 of degree 3 has α_{𝕐/𝕏} = 1. Thus ∑_{P ∈Supp 𝕏,deg_{𝕏}(P)≤1} dim_{K} O_{𝕏,P} = 4, but #𝕏(h)_{≤1} = 1.

In view of this example we shall restict our attention from now on mainly to reduced 0-dimensional subschemes 𝕏 ⊆ ℙ^{2}. The following corollary is an immediate consequence of 2.2 in that case.

2.4. Corollary. Let h be a Castelnuovo function and 𝕏 ⊆ ℙ^{2} a reduced 0-dimensional subscheme with ΔH_{𝕏} = h.

a) We have ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) ≥ ∑_{P ∈Supp 𝕏(h)} deg_{𝕏(h)}(P).

b) We have #𝕏_{σ_{h}+1} ≥ #𝕏(h)_{σ_{h}+1}.

Recall that a 0-dimensional subscheme 𝕏 ⊆ ℙ^{2} is called a Cayley-Bacharach scheme, if every subscheme 𝕐 ⊆ 𝕏 of degree deg 𝕏 − 1 satisfies α_{𝕐/𝕏} = σ_{𝕏} + 1 (cf. [10], 2.1). From 1.9 and 2.4 we get a new proof of the following well-known fact.

2.5. Corollary. Let h be a Castelnuovo function with invariants α_{h} and σ_{h}. The following conditions are equivalent:

a) Every reduced subscheme 𝕏 ⊆ ℙ^{2} with ΔH_{𝕏} = h is a Cayley-Bacharach scheme.

b) We have α_{h} = h_{α_{h}−1} = h_{α_{h}} = … = h_{σ_{h}+1}, i.e. h has the following shape:

−1 0 α_{h} − 1  σ_{h} + 1 α_{h}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  n h_{n}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}

3. Canonical Decompositions

From now on we let 𝕏 ⊆ ℙ^{2} be a reduced 0-dimensional subscheme. It is well-known that if 𝕏 is a Cayley-Bacharach scheme, then its Hilbert function is of decreasing type, i.e. ΔH_{𝕏}(n + 1) < ΔH_{𝕏}(n) for some n ≥ 0 implies ΔH_{𝕏}(m + 1) < ΔH_{𝕏}(m) for all n ≤ m ≤ σ_{𝕏} + 1 (see e.g. [2], 2.5). If ΔH_{𝕏} is not of decreasing type, we are now going to decompose it into “layers”, each of which corresponds to a Castelnuovo function of decreasing type.

0  γ_{ν}τ_{ν}  γ_{2}τ_{2} γ_{1} τ_{1}^{1 α_{1} α_{2} α_{ν}}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  n h_{n}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.} ..... 1^{st} layer 2^{nd} layer ν^{th} layer

3.1. Definition. Let h be a Castelnuovo function with invariants α and σ.

a) If n_{1}, n_{2} ∈ ℕ are such that α < n_{1} < n_{2} and h_{n_{1}−1} > h_{n_{1}} = … = h_{n_{2}} > h_{n_{2}+1}, we call [n_{1}, n_{2}] a flat of h. We also call [α − 1, n_{0}] a flat of h, if n_{0} ≥ α − 1 and α = h_{α−1} = … = h_{n_{0}} > h_{n_{0}+1}. We order the flats by increasing value of h.

Suppose now that h has ν flats.

b) We let τ_{1} = σ + 1 and τ_{i} = max{n | h_{n} > h_{n+1} = h_{n+2} ≥ h_{τ_{i−1}}} for i = 2, …, ν. The number h_{τ_{i+1}} is the last value of h before the i^{th} flat.

c) For i = 0, …, ν − 1 we let α_{i} = h_{τ_{i+1}+1} and α_{ν} = α. The number α_{i} is the value of h at its i^{th} flat.

d) For i = 0, …, ν we let γ_{i} = max{n | h_{n−1} = h_{n} > h_{n+1} ≥ h_{τ_{i}+1}}. This maximum exists, except for the case h_{α−1} > h_{α}, in which we let γ_{ν} = α − 1. The number γ_{i} corresponds to the last time h takes on its value α_{i}.

e) For i = 1, …, ν we let

ΔH_{i}(n) = \{ \begin{matrix}n + 1 & if 0 ≤ n ≤ α_{i} − 1 − α_{i−1}, \\ α_{i} − α_{i−1} & if α_{i} − α_{i−1} ≤ n ≤ γ_{i} − α_{i−1}, \\ h_{n+α_{i−1}} − α_{i−1} & if γ_{i} + 1 − α_{i−1} ≤ n ≤ τ_{i} − α_{i−1}, \\ 0 & otherwise. \end{matrix}

The function ΔH_{i} : ℤ ⟶ ℕ corresponds to the i^{th} layer of h, shifted by α_{i−1} down and to the left.

The importance of this decomposition of Castelnuovo functions comes from the following proposition which follows from the results of [2]. For the convenience of the reader, we include a reduction to those results.

3.2. Proposition. Let 𝕏 ⊆ ℙ^{2} be a reduced 0-dimensional subscheme with Castelnuovo function h = ΔH_{𝕏}. Then there exists a unique decomposition

𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)}

into subschemes 𝕏^{(i)} ⊆ 𝕏 such that ΔH_{𝕏^{(i)}} = ΔH_{i} for i = 1, …, ν, where ν is the number of flats of h and ΔH_{i} is the function defined in 3.1.e.

This decomposition is called the canonical decomposition of 𝕏. The subsets 𝕏^{(i)} ⊆ 𝕏 are called the canonical components of 𝕏. For a function f : ℤ ⟶ ℕ and d ∈ ℕ, we let f^{(d)}(n) := min{d, f(n)} and call f^{(d)} : ℤ ⟶ ℕ the truncation of f at height d.

Proof. To prove existence of the canonical decomposition, we proceed by induction on ν. As the claim is trivially true in case ν = 1, we can assume ν ≥ 2. Then h_{γ_{1}−1} = h_{γ_{1}} ≠ 0 implies by [2], 2.3 that the elements in (I_{𝕏})_{γ_{1}} have a nontrivial gcd F_{1} ∈ K[X_{0}, X_{1}, X_{2}] which is homogeneous of degree α_{1} = ΔH_{𝕏}(γ_{1}). By [2], 2.2.b, we have #(𝕏 ∩ Z(F_{1})) = ∑_{n∈ℤ}(ΔH_{𝕏})^{(α_{1})}(n), so that [2], 2.2.a yields ΔH_{𝕏^{(1)}} = (ΔH_{𝕏})^{(α_{1})} = ΔH_{1} for 𝕏^{(1)} = 𝕏 ∩ Z(F_{1}). By [2], 2.7.b, we have ΔH_{𝕏\𝕏^{(1)}}(n) = ΔH_{𝕏}(n + α_{1}) − (ΔH_{𝕏})^{(α_{1})}(n + α_{1}), and by induction, 𝕏 \ 𝕏^{(1)} has a canonical decomposition 𝕏 \ 𝕏^{(1)} = 𝕏^{(2)}∪· . . . ∪· 𝕏^{(ν)}. Then 𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)} is a canonical decomposition of 𝕏.

Now we prove uniqueness. Suppose 𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)} = 𝕏^{[1]}∪· . . . ∪· 𝕏^{[ν]} are two canonical decompositions of 𝕏. Since ΔH_{𝕏^{(1)}} = ΔH_{𝕏^{[1]}} = ΔH_{1}, the scheme 𝕏^{(1)} is contained in a curve C of degree α_{1}, and 𝕏^{[1]} is contained in a curve C′ of degree α_{1}. By [2], 2.2, it follows that C and C′ are fixed curves of (I_{𝕏})_{α_{1}}, and by [2], 2.3, this fixed curve is unique, i.e. C = C′. Thus 𝕏^{(1)} = 𝕏^{[1]} = 𝕏 ∩ C, and continuing this way we get inductively 𝕏^{(i)} = 𝕏^{[i]} for i = 1, …, ν. □

For i = 1, …, ν, we consider the subschemes 𝕏^{(≥i)} = 𝕏^{(i)}∪· . . . ∪· 𝕏^{(ν)} of 𝕏.

3.3. Corollary. Let i ∈ {1, …, ν}.

a) For all n ∈ ℤ we have ΔH_{𝕏^{(≥i)}}(n) = ΔH_{𝕏}(n + α_{i−1}) − ΔH_{𝕏}^{(α_{i−1})}(n + α_{i−1}).

b) We have σ_{𝕏^{(≥i)}} = τ_{i} − α_{i−1} − 1.

c) For all P ∈ Supp 𝕏^{(i)} we have deg_{𝕏}(P) ≤ τ_{i}.

d) We have ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) ≤ ∑_{i=1}^{ν} τ_{i} · #𝕏^{(i)}.

Proof. Claims a) and b) follow from the proof of 3.2. Claim c) is true in case i = 1, because deg_{𝕏}(P) ≤ τ_{1} = σ_{𝕏} + 1 for all P ∈ Supp 𝕏. In particular, every Q ∈ Supp 𝕏^{(i)} satisfies deg_{𝕏^{(≥i)}}(Q) ≤ τ_{i} − α_{i−1} by b). Since 𝕏 \ 𝕏^{(≥i)} = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(i−1)} is contained in a curve of degree α_{i−1}, it follows that deg_{𝕏}(Q) ≤ (τ_{i} − α_{i−1}) + α_{i−1} = τ_{i}. Claim d) follows by summing up the estimates of c). □

Our next goal is to determine the relation between the degree of a point in 𝕏 and its degree in its canonical component 𝕏^{(i)}. For this we need to generalize [6], 2.7 as follows.

3.4. Proposition. Let 𝕏, 𝕐 ⊆ ℙ^{2} be reduced 0-dimensional subschemes, and let C ⊆ ℙ^{2} be a curve of degree α containing 𝕐, but not containing any point of 𝕏.

a) If n_{0} ∈ ℕ and H_{𝕐}(n) = H_{C}(n) for all n ≤ n_{0}, then

H_{𝕏∪𝕐}(n) = H_{𝕏}(n − α) + H_{𝕐}(n)  for all n ≤ n_{0}.
b) If n_{0} ≥ σ_{𝕏} + 1 + α and H_{𝕐}(n) = H_{C}(n) for all n ≤ n_{0}, then
H_{𝕏∪𝕐}(n) = H_{𝕏}(n − α) + H_{𝕐}(n)  for all n ∈ ℤ.
c) If max{n ∈ ℤ | ΔH_{𝕐}(n) = ΔH_{C}(n)} ≥ σ_{𝕏} + 2 + α_{𝕐}, then
deg_{𝕏∪𝕐}(P) = \{ \begin{matrix}deg_{𝕐}(P) & for all P ∈ Supp 𝕐, \\ deg_{𝕏}(P) + α_{𝕐} & for all P ∈ Supp 𝕏. \end{matrix}

Proof. “a)” Let C = Z(F) for some F ∈ K[X_{0}, X_{1}, X_{2}]_{α}. Since (I_{𝕐})_{n} = (I_{C})_{n} for n ≤ n_{0}, this polynomial F divides all elements of I_{𝕏∪𝕐} = I_{𝕏} ∩I_{𝕐} of degree ≤ n_{0}. In fact, we have (I_{𝕏∪𝕐})_{n} = F ·(I_{𝕏})_{n−α} for n ≤ n_{0}. Thus H_{𝕏∪𝕐}(n) = H_{C}(n)+dim_{K}(I_{C}/I_{𝕏∪𝕐})_{n} = H_{C}(n) + dim_{K}(K[X_{0}, X_{1}, X_{2}]/I_{𝕏})_{n−α} = H_{𝕐}(n) + H_{𝕏}(n − α) for n ≤ n_{0}.

“b)” For n ≤ n_{0}, the claim follows from a). Thus we have H_{𝕏∪𝕐}(n_{0}) = H_{𝕏}(n_{0} − α) + H_{𝕐}(n_{0}) = deg 𝕏 + H_{𝕐}(n_{0}). Now σ_{𝕐} ≤ σ_{𝕏∪𝕐} and ΔH_{𝕐}(n) ≤ ΔH_{𝕏∪𝕐}(n) for all n ∈ ℤ yield

deg 𝕏 + deg 𝕐 = H_{𝕏∪𝕐}(σ_{𝕏∪𝕐} + 1) = H_{𝕏∪𝕐}(n_{0}) + ∑_{n=n_{0}+1}^{σ_{𝕏∪𝕐}+1}ΔH_{𝕏∪𝕐}(n)
≥ deg 𝕏 + H_{𝕐}(n_{0}) + ∑_{n=n_{0}+1}^{σ_{𝕐}+1}ΔH_{𝕐}(n) = deg 𝕏 + deg 𝕐,

and therefore σ_{𝕐} = σ_{𝕏∪𝕐} as well as ΔH_{𝕐}(n) = ΔH_{𝕏∪𝕐}(n) for n_{0} + 1 ≤ n ≤ σ_{𝕏∪𝕐} + 1. Since H_{𝕏}(n − α) = deg 𝕏 for n ≥ n_{0}, the claim follows.

“c)” Let m = max{n ∈ ℤ | ΔH_{𝕐}(n) = ΔH_{C}(n)}, let P ∈ Supp 𝕐, and let 𝕐′ = 𝕐 \ {P }. From 1.9 and 2.2 we know that α_{𝕐′/𝕐} = deg_{𝕐}(P) ≥ m. By applying b) for 𝕐′ instead of 𝕐 and n_{0} = m − 1, we calculate

ΔH_{𝕏∪𝕐′}(n) = ΔH_{𝕏}(n − α_{𝕐}) + ΔH_{𝕐′}(n)

= ΔH_{𝕏}(n − α_{𝕐}) + ΔH_{𝕐}(n) + \{ \begin{matrix}0 & if n ≠ α_{𝕐′/𝕐}, \\ −1 & if n = α_{𝕐′/𝕐}. \end{matrix}

Therefore we get deg_{𝕏∪𝕐}(P) = α_{𝕐′/𝕐} = deg_{𝕐}(P). In case P ∈ Supp 𝕏, we let 𝕏′ = 𝕏 \ {P } and apply b) in the analogous manner. Then

ΔH_{𝕏′∪𝕐}(n) = ΔH_{𝕐}(n) + ΔH_{𝕏′}(n − α_{𝕐})

= ΔH_{𝕐}(n) + ΔH_{𝕏}(n − α_{𝕐}) + \{ \begin{matrix}0 & if n ≠ α_{𝕏′/𝕏} + α_{𝕐}, \\ −1 & if n = α_{𝕏′/𝕏} + α_{𝕐} \end{matrix}

yields deg_{𝕏∪𝕐}(P) = α_{𝕏′/𝕏} + α_{𝕐} = deg_{𝕏}(P) + α_{𝕐}. □

Now we are ready to prove the desired formula.

3.5. Proposition. Let 𝕏 ⊆ ℙ^{2} be a reduced 0-dimensional subscheme with canonical decomposition 𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)}. Then for every i ∈ {1, …, ν} and P ∈ Supp 𝕏^{(i)} we have

deg_{𝕏}(P) = deg_{𝕏^{(i)}}(P) + α_{i−1}.

Proof. This is shown by induction on ν. The case ν = 1 being clear, we can assume that ν > 1. By the construction given in the proof of 3.2, we have 𝕏^{(1)} = 𝕏 ∩ Z(F_{1}), ..., 𝕏^{(ν−1)} = (𝕏 \ ⋃_{i=1}^{ν−2} 𝕏^{(i)}) ∩ Z(F_{ν−1}), and F = F_{1} ⋯ F_{ν−1} defines a curve of degree α_{ν−1} containing 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν−1)}, but not containing any point of 𝕏^{(ν)}. Now we apply 3.4 with 𝕏^{(ν)} instead of 𝕏, with 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν−1)} instead of 𝕐, and with C = Z(F), and we get

deg_{𝕏}(P) = \{ \begin{matrix}deg_{𝕏^{(1)}∪· ...∪· 𝕏^{(ν−1)}}(P) & if P ∈ Supp (𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν−1)}), \\ deg_{𝕏^{(ν)}}(P) + α_{ν−1} & if P ∈ Supp 𝕏^{(ν)}. \end{matrix}

Using the induction hypothesis finishes the proof. □

Recall that, by [2], 4.4, for every Hilbert function of decreasing type there exists a 0-dimensional reduced Cayley-Bacharach scheme 𝕏 ⊆ ℙ^{2} having that Hilbert function. In case char K = 0, this follows also from Harris’ Uniform Proposition Lemma (cf. [7], p. 197). Given an arbitrary Castelnuovo function, we shall now construct a 0-dimensional reduced subscheme 𝕏 ⊆ ℙ^{2} having that Castelnuovo function and canonical components which are Cayley-Bacharach schemes.

3.6. Proposition. Let h : ℤ ⟶ ℕ be a Castelnuovo function with ν flats, and let ΔH_{i} be the corresponding functions defined in 3.1.e. For each i ∈ {1, …, ν}, we choose a reduced Cayley-Bacharach scheme 𝕐_{i} having Castelnuovo function ΔH_{𝕐_{i}} = ΔH_{i}.

a) There are projective coordinate transformations T_{i} : ℙ^{2} ⟶^{∼} ℙ^{2} for i = 1, …, ν such that the scheme 𝕏 = T_{1}(𝕐_{1})∪· . . . ∪· T_{ν}(𝕐_{ν}) has Castelnuovo function ΔH_{𝕏} = h.

b) In the canonical decomposition 𝕏 = 𝕏^{(1)}∪· . . . ∪· 𝕏^{(ν)} of this scheme 𝕏 we have 𝕏^{(i)} = T_{i}(𝕐_{i}) for i = 1, …, ν.

c) For i ∈ {1, …, ν} and P ∈ Supp 𝕏^{(i)} we have deg_{𝕏}(P) = τ_{i}.

d) We have ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) = ∑_{i=1}^{ν} τ_{i} · deg 𝕏^{(i)}. In particular, the bound of Corollary 3.3.d is sharp.

Proof. “a)” Since in case ν = 1 we can choose T_{1} = id_{ℙ^{2}}, we may assume ν > 1. For i ∈ {1, …, ν − 1}, we let m_{i} = max{n ≥ α_{𝕐_{i}} | ΔH_{i}(n) = α_{𝕐_{i}}}, and we observe that m_{i} = γ_{i} − α_{i−1} ≥ τ_{i+1} + 2 − α_{i−1} = σ_{𝕐_{i+1}} + 3 + α_{i} − α_{i−1} = σ_{𝕐_{i+1}} + 3 + α_{𝕐_{i}}. Let C_{ν−1} be a curve of degree α_{𝕐_{ν−1}} containing 𝕐_{ν−1}. We let T_{ν} = id_{ℙ^{2}} and we choose T_{ν−1} : ℙ^{2} ⟶^{∼} ℙ^{2} such that T_{ν−1}(C_{ν−1}) does not contain any point of 𝕐_{ν}. Because of our observation above, we can apply 3.4.b with 𝕐_{ν} instead of 𝕏, with T_{ν−1}(𝕐_{ν−1}) instead of 𝕐, with C = T_{ν−1}(C_{ν−1}), and with n_{0} = m_{ν−1}. We obtain

ΔH_{T_{ν−1}(𝕐_{ν−1})∪T_{ν}(𝕐_{ν})}(n) = ΔH_{𝕐_{ν}} (n − α_{𝕐_{ν−1}}) + ΔH_{𝕐_{ν−1}}(n)

for all n ∈ ℤ. Next we choose a curve C_{ν−2} of degree α_{𝕐_{ν−2}} containing 𝕐_{ν−2}, and T_{ν−2} : ℙ^{2} ⟶^{∼} ℙ^{2} such that T_{ν−2}(C_{ν−2}) does not contain any point of T_{ν−1}(𝕐_{ν−1})∪· T_{ν}(𝕐_{ν}).

By applying 3.4.b in the analogous manner and continuing this procedure, we find projective coordinate transformations T_{1}, …, T_{ν} such that 𝕏 = T_{1}(𝕐_{1})∪· . . . ∪· T_{ν}(𝕐_{ν}) has Castelnuovo function

\begin{aligned}ΔH_{𝕏}(n) & = ΔH_{𝕐_{ν}} (n − α_{𝕐_{ν−1}} − … − α_{𝕐_{1}}) + … + ΔH_{𝕐_{1}}(n) \\ & = ΔH_{ν}(n − α_{ν−1}) + … + ΔH_{1}(n − α_{0}) = h_{n}. \end{aligned}

Claims b), c) and d) follow from 3.2 and 3.5. □

4. Schemes with extremal conductor ideals

Given a reduced 0-dimensional subscheme 𝕏 ⊆ ℙ^{2}, the projective coordinate ring R = K[X_{0}, X_{1}, X_{2}]/I_{𝕏} is a 1-dimensional graded Cohen-Macaulay ring with homogeneous maximal ideal 𝔪 = ⊕_{n≥1}R_{n} and a linear nonzerodivisor x_{0} = X_{0} + I_{𝕏} ∈ R_{1}. Let Supp 𝕏 = {P_{1}, …, P_{s}} and P_{i} = (1 : p_{i} : q_{i}) for i = 1, …, s. The homogeneous ideal of {P_{i}} in 𝕏 is the image 𝔭_{i} of 𝔓_{i} = (X_{1} − p_{i}X_{0}, X_{2} − q_{i}X_{0}) in R. We start this section with an explicit computation of the full ring of quotients Q(R) and the integral closure \overline{R} of R in Q(R).

4.1. Proposition. a) We have Q(R) ≅ ∏_{i=1}^{s} K(T_{i}), and the canonical map R ⟶ Q(R) is given by

\begin{matrix}R_{n} & ⟶ ∏_{i=1}^{s}K(T_{i}) \\ r & ⟼ (r(P_{1})T_{1}^{n}, …, r(P_{s})T_{s}^{n}) \end{matrix}

for all n ≥ 0. Here we let r(P_{i}) = r(1, p_{i}, q_{i}) for i = 1, …, s.

b) We have \overline{R} ≅ ∏_{i=1}^{s} K[T_{i}], and the canonical map R ⟶ \overline{R} is induced by the map R ⟶ Q(R).

Proof. Since 𝕏 is a reduced scheme, we have R/𝔭_{i} ≅ K[T_{i}] with T_{i} = x_{0} + 𝔭_{i} for i = 1, …, s, and we have 𝔭_{1} ∩ … ∩ 𝔭_{s} = 0. Thus the Chinese Remainder Theorem yields an injection R ⟶ ∏_{i=1}^{s} K[T_{i}] which is given by r ⟼ (r(P_{1})T_{1}^{n}, …, r(P_{s})T_{s}^{n}) for r ∈ R_{n}. For dimension reasons, this map is an isomorphism in degrees n ≥ σ_{𝕏} + 1, and therefore ∏_{i=1}^{s} K[T_{i}, T_{i}^{−1}] ≅ R_{x_{0}}. From this claims a) and b) follow immediately.

□

4.2. Definition. a) We call 𝔉 := \{r ∈ R | r · \overline{R} ⊆ R\} ⊆ R the conductor ideal of 𝕏. b) The number ℓ(R/𝔉) is called the conductor colength of 𝕏.

Notice that ℓ(R/𝔉) is finite, since r · \overline{R} ⊆ R for r ∈ R_{n} and n ≥ σ_{𝕏} + 1 (cf. 4.1). Recall that the following descriptions of 𝔉 were proven in [5], 3.13.

4.3. Proposition. a) In \overline{R}, the ideal 𝔉 is given by 𝔉 = (T_{1}^{deg_{𝕏}(P_{1})}, …, T_{s}^{deg_{𝕏}(P_{s})}) · \overline{R}.

b) In R, the ideal 𝔉 is given by 𝔉 = (f_{1}, …, f_{s}), where f_{i} is a minimal separator of 𝕏 \ {P } in 𝕏.

From those descriptions we immediately get formulas for ℓ(\overline{R}/𝔉) and ℓ(R/𝔉).

4.4. Corollary. a) We have ℓ(\overline{R}/𝔉) = ∑_{P ∈Supp 𝕏} deg_{𝕏}(P).

b) We have ℓ(R/𝔉) = ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) − ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i).

Proof. By 4.1.b and 4.3.a, the ring \overline{R}/𝔉 is isomorphic to ∏_{i=1}^{s} K[T_{i}]/(T_{i}^{deg_{𝕏}(P_{i})}), and this yields formula a). Formula b) follows from a) and the exact sequence

0 ⟶ R/𝔉 ⟶ \overline{R}/𝔉 ⟶ \overline{R}/R ⟶ 0,

since we can calculate ℓ(\overline{R}/R) = ∑_{n=0}^{∞}(dim_{K} \overline{R}_{n} − dim_{K} R_{n}) = ∑_{n=0}^{σ_{𝕏}}(s − H_{𝕏}(n)) = ∑_{n=0}^{σ_{𝕏}} ∑_{i=n+1}^{σ_{𝕏}+1} ΔH_{𝕏}(i) = ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i). □

Using the results of the previous sections, we are able to give sharp estimates for ℓ(R/𝔉) which depend only on the Castelnuovo function of 𝕏.

4.5. Proposition. Let 𝕏 ⊆ ℙ^{2} be a reduced 0-dimensional subscheme with Castelnuovo function h = ΔH_{𝕏}, and let 𝕏(h) be some Castelnuovo set for h.

a) We have ℓ(R/𝔉) ≥ ∑_{P ∈Supp 𝕏(h)} deg_{𝕏(h)}(P)−∑_{i=0}^{σ_{𝕏}+1} i·ΔH_{𝕏}(i). The right-hand side of this inequality depends only on h. For any Castelnuovo set for h it is an equality.

b) We have ℓ(R/𝔉) ≤ ∑_{i=1}^{ν} τ_{i}·deg 𝕏^{(i)}−∑_{i=0}^{σ_{𝕏}+1} i·ΔH_{𝕏}(i). Here ν and τ_{i} are defined as in 3.1, and 𝕏^{(i)} are the canonical components of 𝕏. Again the right-hand side depends only on h, and there exists a reduced 0-dimensional subscheme of ℙ^{2} achieving equality.

Proof. “a)” Using 2.4.a and 4.4, we have ℓ(R/𝔉) = ∑_{P ∈Supp (𝕏)} deg_{𝕏}(P) − ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i) ≥ ∑_{P ∈Supp 𝕏(h)} deg_{𝕏(h)}(P) − ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i) = ℓ(R_{𝕏(h)}/𝔉_{𝕏(h)}), i.e. the conductor colength is minimal for Castelnuovo sets among all reduced 0-dimensional schemes having Castelnuovo function h. The independence of the right-hand side of the choice of 𝕏(h) follows from 1.9.

“b)” The inequality follows from 4.4.b and 3.3.d. Schemes achieving equality have been constructed in 3.6.d. □

The remainder of this section will be devoted to relating conductor colength with the following invariants.

4.6. Definition. a) The number ℓ(\overline{R}/R) is called the singularity degree of 𝕏. b) The number δ_{𝕏} = ℓ(\overline{R}/R) − ℓ(R/𝔉) is called the superabundance of 𝕏.

4.7. Remarks. a) The affine cone C over 𝕏 consists of s reduced lines in 𝔸^{3} meeting in the origin O = (0, 0, 0). For the local ring of C at O we have O_{C,O} = R_{𝔪}. Since the singularity degree of 𝕏 satisfies ℓ(\overline{R}/R) = ℓ(\overline{R}_{𝔪}/R_{𝔪}), it is nothing but the usual singularity degree of C at O. As we shall see below, we always have ℓ(\overline{R}/R) ≥ ℓ(R/𝔉) which explains the term “superabundance” for δ_{𝕏}.

b) It follows from the exact sequence 0 ⟶ R/𝔉 ⟶ \overline{R}/𝔉 ⟶ \overline{R}/R ⟶ 0 that

δ_{𝕏} = ℓ(\overline{R}/𝔉) − 2ℓ(R/𝔉).

Below we shall prove a refinement of the inequality δ_{𝕏} = ℓ(\overline{R}/𝔉) − 2ℓ(R/𝔉) ≥ 0 which was shown in [9], 3.7, for the local situation.

c) Using 4.4.b, we can express δ_{𝕏} as δ_{𝕏} = 2 ∑_{i=0}^{σ_{𝕏}+1} i·ΔH_{𝕏}(i)−∑_{P ∈Supp 𝕏} deg_{𝕏}(P).

Our next proposition yields sharp inequalities for the superabundance of 𝕏.

4.8. Proposition. Let 𝕏 ⊆ ℙ^{2} be a reduced 0-dimensional subscheme with Castelnuovo function h = ΔH_{𝕏}. Using the notations of 3.1, we define the quantity

δ′_{h} = 2 ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i) − ∑_{i=1}^{ν} τ_{i} · deg 𝕏^{(i)}

which only depends on h. Let 𝕏(h) be any Castelnuovo set for h.

a) We have the chain of inequalities

δ_{𝕏(h)} ≥ δ_{𝕏} ≥ δ′_{h} ≥ 0.

Here the quantities δ_{𝕏(h)} and δ′_{h} only depend on h.

b) In the above chain we have δ′_{h} = 0 if and only if ΔH_{𝕏} is symmetric, i.e. if and only if ΔH_{𝕏}(σ_{𝕏} + 1 − n) = ΔH_{𝕏}(n) for all n ∈ ℤ.

c) In the above chain of inequalities we have δ_{𝕏} = δ′_{h} if and only if 𝕏^{(i)} is a Cayley-Bacharach scheme for each i ∈ {1, …, ν}.

d) In the above chain of inequalities we have δ_{𝕏} = δ′_{h} = 0 if and only if 𝕏 is arithmetically Gorenstein, i.e. iff R is a Gorenstein ring. In other words, we have 2ℓ(R/𝔉) ≤ ℓ(\overline{R}/𝔉), with equality if and only if 𝕏 is arithmetically Gorenstein.

Proof. “a)” The claim δ_{𝕏} ≤ δ_{𝕏(h)} follows from 2.4.a and 4.7.c. The claim δ_{𝕏} ≥ δ′_{h} follows from 3.3.d and 4.7.c. Thus we have to show δ′_{h} ≥ 0. For i = 0, …, α_{𝕏} − 1 we let e_{i} = σ_{𝕏} + 1 − max{n ≥ α_{𝕏} | h_{n} > i}.

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Then we have 0 = e_{0} ≤ e_{1} ≤ … ≤ e_{α_{𝕏}−1} ≤ σ_{𝕏} + 1 − α_{𝕏}, and using the notations of 3.1, a straightforward calculation yields

δ′_{h} = ∑_{i=1}^{ν} ∑_{j=α_{i−1}}^{α_{i}−1}(σ_{𝕏} + 2 − e_{j} − j)(σ_{𝕏} + 1 + j − e_{j} − τ_{i}).

Similarly, starting from 𝕏^{(i)} we define numbers e_{j}^{(i)} = σ_{𝕏^{(i)}} + 1 − max{n ≥ α_{𝕏^{(i)}} | ΔH_{i}(n) > j} for all i = 1, …, ν and j = 0, …, α_{i} − α_{i−1} − 1. Clearly, we have e_{j}^{(i)} = e_{j+α_{i−1}} − (σ_{𝕏} + 1) + τ_{i}, and another straightforward calculation yields

δ′_{h} = ∑_{i=1}^{ν} ∑_{j=0}^{α_{i}−α_{i−1}−1}(τ_{i} + 1 + e_{j}^{(i)} − j − α_{i−1}) · (j − e_{j}^{(i)}) + ∑_{i=1}^{ν} α_{i−1} · deg 𝕏^{(i)}.

Since τ_{i} + 1 + e_{j}^{(i)} − j − α_{i−1} = σ_{𝕏} + 2 − e_{j+α_{i−1}} − (j + α_{i−1}) is the number of points of 𝕏(h) contained in the line Z(X_{2} − t_{j+α_{i−1}}X_{0}), and since the fact that ΔH_{i} is of decreasing type implies e_{j}^{(i)} ≤ j, we conclude that δ′_{h} ≥ 0.

“b)” From the formula above we find that δ′_{h} = 0 if and only if ν = 1 and e_{j}^{(1)} = j for j = 0, …, α_{𝕏} − 1. This in turn is equivalent to ΔH_{𝕏} being symmetric.

“c)” We have δ_{𝕏} = δ′_{h} if and only if ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) = ∑_{i=1}^{ν} τ_{i} · deg 𝕏^{(i)}. In view of deg_{𝕏}(P) = deg_{𝕏^{(i)}}(P) + α_{i−1} ≤ σ_{𝕏^{(i)}} + 1 + α_{i−1} ≤ σ_{𝕏^{(≥i)}} + 1 + α_{i−1} = τ_{i} for P ∈ Supp 𝕏^{(i)}, this happens if and only if each 𝕏^{(i)} is a Cayley-Bacharach scheme for i = 1, …, ν.

“d)” This claim follows immediately from b), c), and [11], 1.1. □

Using the numbers e_{i} = σ_{𝕏} + 1 − max{n ≥ α_{𝕏} | h_{n} > i} defined in the proof of this proposition, we can make the first inequality of 4.8.a even more explicit.

4.9. Corollary. We have δ_{𝕏(h)} = \frac{1}{6}α_{𝕏}(α_{𝕏} − 1)(3σ_{𝕏} − 2α_{𝕏} + 7) − ∑_{j=0}^{α_{𝕏}−1} je_{j} ≥ δ_{𝕏}.

Proof. Each of the points of 𝕏(h) contained in Z(X_{2} − t_{j}X_{0}) has degree σ_{𝕏} + 1 − e_{j}. Plugging this into δ_{𝕏(h)} = 2 ∑_{i=0}^{σ_{𝕏}+1} i · ΔH_{𝕏}(i) − ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) = ∑_{j=0}^{α_{𝕏}−1}(σ_{𝕏} + 2 − e_{j} − j)(σ_{𝕏} + 1 + j − e_{j}) − ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) yields the desired result after a short calculation. □

Our final result generalizes 2.5 and characterizes Castelnuovo functions h such that every reduced 0-dimensional scheme 𝕏 having ΔH_{𝕏} = h satisfies δ_{𝕏} = δ′_{h}.

4.10. Proposition. For a Castelnuovo function h, the following statements are equivalent.

a) Every reduced 0-dimensional subscheme 𝕏 ⊆ ℙ^{2} with ΔH_{𝕏} = h satisfies δ_{𝕏} = δ′_{h}.

b) In the decomposition of h (cf. 3.1) we have τ_{i} = γ_{i} for all i ∈ {1, …, ν}.

Proof. Suppose condition a) is satisfied. In particular, this implies that δ_{𝕏(h)} = δ′_{h}, i.e. that ∑_{P ∈Supp 𝕏(h)} deg_{𝕏(h)}(P) = ∑_{i=1}^{ν} τ_{i} · deg 𝕏^{(i)}. As in the proof of 4.8.c, it follows that each 𝕏^{(i)} is a Cayley-Bacharach scheme. Thus ΔH_{i} is of the type described in 2.5, i.e. we have τ_{i} = γ_{i}.

Conversely, suppose that γ_{i} = τ_{i} for i = 1, …, ν. Then 2.5 implies that each 𝕏^{(i)} is a Cayley-Bacharach scheme. Thus ∑_{P ∈Supp 𝕏} deg_{𝕏}(P) = ∑_{i=1}^{ν} τ_{i} · deg 𝕏^{(i)}, and the claim follows from 4.7.c and the definition of δ′_{h}. □

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Address of the Authors

Fakultät für Mathematik Universität Regensburg Postfach 397 D-93040 Regensburg Germany