High order vanishing ideals at n + 3 points of P^{n}

Clare D’Cruz Chennai Mathematical Institute 92, G. N. Chetty Road Madras 600017, INDIA clare@smi.ernet.in

Anthony Iarrobino Department of Mathematics, 567 Lake Hall Northeastern University Boston, Massachusetts 02115, USA iarrobin@neu.edu

October 20, 1998, revised May 21, 1999

We discuss a particular instance of an open problem concerning the Hilbert function of vanishing ideals at points of P^{n}, and we discuss some related results, and methods of approach to the special case. The general problem is discussed in [I1, I2]. We choose to restrict ourselves here to the equal vanishing orders problem of HP OINT S, or equal degrees case of HP OW LIN. It is our hope that a concise discussion of a quite special case might illustrate the connection between the Algebraic and Geometric problem below, and encourage others to work on them. The authors thank the Catania organizers for their warm hospitality. We thank A. Geramita for pointing out to us the preprint of M. V. Catalisano, P. Ellia, and A. Gimigliano [CEG], and we thank M. V. Catalisano for a helpful discussion.

I. The Algebraic and Geometric Problems. Let R = k[x_{1}, … x_{r}] be the polynomial ring and denote by HP OW LIN(s, j, r) (for power of linear) the Hilbert function H(A) for the Artin ring A = A_{L}(j) = R/L^{j} = R/(L_{1}^{j}, … L_{s}^{j}), defined by the j-th powers of a set L of s generic linear homogeneous elements of R. Thus, H(A)_{i} = dim_{k}(R_{i}/(L^{j})_{i}). Here, L^{j} has a symbolic meaning, and is not the j-th power of the ideal L. We assume henceforth s > r + 1, since the complete intersection (s ≤ r) and almost complete intersection (s = r + 1) Hilbert functions are known (see [I1§1.1]). The general Algebraic Problem is

PROBLEM A. Determine HP OW LIN(s, j, r) for all triples (s, j, r).

This problem is equivalent to the following problem for vanishing ideals at points of P^{n}, n = r − 1. Let HP OINT S(s, a, n) denote the Hilbert function H(R/I_{P}^{[a]}) of the a-th symbolic power of the vanishing ideal I_{P} at s generic points of P^{n}. The general Geometric Problem is

PROBLEM B. Determine HP OINT S(s, a, n) for all triples (s, a, n).

These problems have been studied by many. In particular there was a classical study of Problem 2 for a = 2 because of its connection to the Waring problem for forms; this case was recently solved by J. Alexander and A. Hirschowitz; K. Chandler has also studied the case a = 2 and as well the case a = 3 (see [T],[AH], [Ch]). We denote by HGEN(s, j, r) (for generic ) and HF ROBERG(s, j, r), respectively, the actual and the conjectural Hilbert function of B_{F} = R/F (s, j) = R/(f_{1}, … f_{s}), where F (s, j) = {f_{1}, …, f_{s}} is a general enough set of degree-j homogeneous elements of R. The Fröberg series

HF ROBERG(s, j, r)(Z) = ∑ HF ROBERG(s, j, r)_{i}Z^{i}

satisfies

HF ROBERG(s, j, r)(Z) = | (1 − Z)^{−r} · (1 − Z^{j})^{s} |  (1)

where | S(Z) | is the series obtained by replacing any terms that follow the first negative or zero term of S(Z) by zero (see [F],[FH],[An],[I1]). The Strong Fröberg Conjecture (SFC) is that HGEN(s, j, r) = HF ROBERG(s, j, r); the Weak Fröberg Conjecture (WFC) is that HGEN(s, j, r) ≥ HF ROBERG(s, j, r) termwise [F]. These Froberg Conjectures are widely believed to be true, but have been resistent so far to a general proof. For special cases see Fröberg’s discussion (ibid.), D. Anick’s proof of SFC for r = 3 [An], ^{1} and M. Aubry’s study of SFC in [Au]; for a discussion of their status, see [I1(1.1)].

Since the ideal L^{j} is more special than F (s, j), we have, evidently,

HP OW LIN(s, j, r) ≥ HGEN(s, j, r),  (2)

In most cases where s ≥ r + 4, the second author has conjectured that there is equality in (2), with a finite number of exceptional triples (see [I1]). However, the case s = r + 2 was identified as one of several where in general there is inequality in (2): the Hilbert function HP OW LIN(r+2, j, r) is different from HGEN(r+2, j, r). This is certainly related to the fact, r+2 points in P^{r−1} lie on a rational normal curve.

1But ignore the incorrect statement in [An] that the Weak Fröberg inequality is known

The first author, using calculations in Macaulay and Macaulay II at Northeastern University, made a striking experimental observation for even r = 2t, t ≤ 7. Observation 1 (Conjecture for t > 7). If r = 2t, then

HP OW LIN(r + 2, 2, r)_{t} = 2^{t}.  (3)

For odd r = 2t − 1, the corresponding observation and conjecture (see [I2]) is

HP OW LIN(r + 2, 2, r)_{t} = 1.  (4)

Problem, Special case. Prove the Observation and Conjecture 1 above, using algebraic or geometric methods. In Proposition 7 we will show a rather weaker statement, that HP OW LIN(r + 2, 2, 2t)_{t} ≥ t + 1.

Recall that the socle degree of the Hilbert function H = (1, …, h_{i}, …) of an Artinian algebra is SocDeg(H) = max{i | h_{i} ≠ 0}. Lemmas 2 and 3 below imply that the Observation above concerns the socle degree of HP OW LIN(r + 2, 2, r). Lemma 2 [I2, Lemma 6.2]. If r = 2t − 1, and j ≥ 1, the socle degree of HP OW LIN(r + 2, j, r) is (j − 1)t. Thus, HP OW LIN(r + 2, j, r)_{(j−1)t} ≠ 0 and HP OW LIN(r + 2, j, r)_{(j−1)t+1} = 0. Lemma 3 [C. D’Cruz]. If r = 2t the socle degree of HP OW LIN(r + 2, j, r) is (j − 1)t for all j ≥ 1. Proof: The proof of Lemma 3 is partly based on Lemma 2. Let j ≥ 2 and let I denote the ideal generated by L_{1}^{j}, …, L_{s}^{j}. We may assume that L_{i} = x_{i} for all i = 1, …, r. The exact sequence

0 ⟶ (I : x_{r}^{j−1})_{i} ⟶ R_{i} ⟶^{x_{r}^{j−1}} (\frac{R}{I})_{i+j−1} ⟶ (\frac{R}{(I, x_{r}^{j−1})})_{i+j−1} ⟶ 0

gives

dim_{k} (\frac{R}{I})_{i+j−1} = dim_{k} (\frac{R}{I, x_{r}^{j−1}} )_{i+j−1} + dim_{k} (\frac{R}{I : x_{r}^{j−1}} )_{i}
≤ dim_{k} (\frac{R}{I, x_{r}^{j−1}} )_{i+j−1} + dim_{k} (\frac{R}{I, x_{r}} )_{i} since (I, x_{r}) ⊆ (I : x_{r}^{j−1})
≤ dim_{k} (\frac{R}{I, x_{r}^{j−2}} )_{i+j−1} + dim_{k} (\frac{R}{I, x_{r}} )_{i+1} + dim_{k} (\frac{R}{I, x_{r}} )_{i}
≤ ⋮
≤ ∑_{n=0}^{j−1} dim_{k} (\frac{R}{I, x_{r}} )_{i+n}

Now R/x_{r} ≅ k[x_{1}, …, x_{r−1}] and SOCDEG(r + 1, j, r − 1) is (j − 1)t, by Lemma 2. Hence SOCDEG(r + 2, j, r) is at least (j − 1)t and at most (j − 1)(t + 1). By a result of R. Stanley,

HP OW LIN(r + 1, j, r)_{(j−1)t+1} = HF ROBERG(r + 1, j, r)_{(j−1)t+1} = 0

(see [I1, p. 327, Lemma C]). This implies that HP OW LIN(r + 2, j, r)_{(j−1)t+n} = 0 for all n ≥ 1.

II. Approaches to the Special Problem There are several approaches we are investigating, none of which is complete. For a statement of what was known about the general Algebraic/Geometric Problem as of 1997 see [I1].

II A. An Algebraic approach: specifying L_{1}, …, L_{r+2}.

Consider the following choice of the linear forms {L_{i}}, i = 1, …, r + 2. We let L_{i} = x_{i} for i ≤ r, and set

L_{r+1} = ∑ x_{i} , L_{r+2} = ∑ c_{i}x_{i}  (5)

for general enough coefficients c_{i} ∈ k. Then it is well known that if A = R/L^{j} (i.e. R/(L_{1}^{j}, …, L_{r+2}^{j})) for this choice, then HP OW LIN(r + 2, j, r) = H(A). Now, considering j = 2, we let V denote the quotient ring

V = V (r) = R/(x_{1}^{2}, … x_{r}^{2}).  (6)

Then,

A = V/(f, g), f = ∑_{i<j} x_{i}x_{j}, g = ∑_{i<j} c_{i}c_{j}x_{i}x_{j}.  (7)

Let W (r) denote the following subspace of V (r)_{t}, of dimension 2^{t}.

W (r) = ⟨x_{1}, x_{2}⟩ · ⟨x_{3}, x_{4}⟩ · . . . ⟨x_{r−1}, x_{r}⟩ ⊂ V (r)_{t}.  (8)

Note that W (r) is spanned by the set of all possible products, one variable from each of the t-sets of two variables. Thus W (4) = ⟨x_{1}x_{3}, x_{1}x_{4}, x_{2}x_{3}, x_{2}x_{4}⟩, and x_{1}x_{4}x_{5} ∈ W (6). Clare D’Cruz proposes

Conjecture 4 . W (r) is a vector space complement to (f, g)_{t} in V (r)_{t}.

We have checked Conjecture 4 algebraically up to r = 6, and computationally up to r = 10. The algebraic verification of case r = 6 involves showing a 12 × 12 matrix with coefficients in k[C] = k[c_{1}, … c_{6}] has rank 12. A similar verification for r = 8 would require showing a 56 × 54 matrix has maximal rank, so is a step up.

IIB. Example: the case r = 8. The first case not shown algebraically is r = 8, where by computer calculation,

HP OW LIN(10, 2, 8)_{4} = 16 ≠ 15 = HF ROBERG(10, 2, 8)_{4}.

Here are several approaches that might be attempted in this case.

i. Find the nontrivial relation af + bg = 0, a, b ∈ V (2) between f, g in V_{4}; that is, determine a = A(c), b = B(c) in terms of the variables c_{1}, … c_{8}, which are the coefficients of g. There is of course the trivial relation fg − gf = 0; but computer calculation gives HP OW LIN(10, 2, 8)_{4} = 16 and some arithmetic shows there is an extra relation. Note that such an extra relation is possible between only two generators f, g, since we are working over the quotient ring V.

Finding the extra relation is sufficient to obtain an algebraic proof of this case. It involves 54 equations in 55 unknowns; the relevent matrix has entries that are up to quadratic in the c’s. It is likely that special cases can be explored by computer algebra, as a guide to obtaining algebraic solutions for a(c), b(c).

ii. Find the 16 dimensional annihilating space D(8) of (f, g)_{4} in V (8)_{4}, under the natural Macaulay duality. Using a known construction for r = 7, we can find an 8-dimensional subspace of D(8). This approach is potentially more difficult than resolving Conjecture 4 above in this case (r = 8), since here we ask for an annihilator space in the dual space, rather than just a complement. This approach is related to following geometric construction (iii) and to that of Lemma 7 below; for reasons of space we don’t describe it further.

iii. The related Geometric Problem – from Macaulay duality.

Consider a subset P of projective space P^{7} consisting of ten points, first the usual coordinate points, e_{1}, …, e_{8}, then (1, …, 1), and (c_{1}, … c_{8}), with the latter point sufficiently general. Let I_{P}^{[a]} denote the a-th symbolic power of the defining ideal I_{P}: it is the ideal in R of functions vanishing to order at least a, (the function and all its order 1, …, a − 1 partial derivatives vanish at P). Using Macaulay duality, as in [EmI] and [I1], finding D(8) is equivalent to finding the dimension-16 vector space (I_{P}^{[3]})_{4} of quartics vanishing to order 3 on these 10 points of projective space P^{7}. A nine dimensional subspace is given by the construction in Lemma 7 below.

IIC. The Geometric setting: r + 2 points on a rational normal curve in P^{r−1}.

We now describe further the geometric context for the special problem. Recall that HP OINT S(s, a, n) denotes the Hilbert function H(R/I_{P}^{[a]}) of the a-th symbolic power of the vanishing ideal I_{P} at s generic points of P^{n}. Then (3) in Observation-Conjecture 1 above is equivalent to the “dual” observation and problem, to show

HP OINT S(r + 2, t − 1, r − 1)_{t} = dim_{k}(R_{t}) − 2^{t}.  (9)

Equivalently, this is to show dim_{k}(I_{P}^{[t−1]}_{t}) = 2^{t}, where P is the set of r + 2 points e_{1}, …, e_{r}, (1, …, 1), and (c_{1}, …, c_{r}) on P^{r−1}, with the last point general.

It is well known that r + 2 general enough points on P^{r−1} determine a unique rational normal curve on P^{r−1} (see Thm 1.18 of [Ha]). The article of Catalisano, Ellia, and Gimigliano, [CEG] studies geometrically the Hilbert function HP OINT S(s, A, n)_{i} of order-A = (a_{1}, …, a_{s})-vanishing ideals at a sequence of s points lying on a rational normal curve in P^{n}. Their main result Theorem 3.2 requires degree i at least 2a − 1, a = Max(a_{i}) and corresponds to pairs (j, i) in HP OW LIN(s, j, r)_{i} where i ≤ 2j − 1. Note that the degree i = 2j is when the first Koszul relations occur for the ideal L^{j}; also the pairs (j, i) for which i ≤ 2j − 1 are those where the Weak Fröberg Conjecture (WFC) is known, and when WFC entails only the well known multiplicity bound for HP OINT S(s, a, n). (The WFC bound is far from determining HP OW LIN(s, j, r)).

The case (j, i) = (2, t) of HP OW LIN(r + 2, j, r)_{i} whose study we propose here is at the opposite extreme - it is at the socle degree of HP OW LIN(r+2, 2, r) where the highest order Koszul relations among the r+2 generators of L^{2} affect the Hilbert function – the Koszul order is about t/2−1 ≅ r/4. Geometrically, our problem from Equation (9) is to study the vanishing ideal I_{P}^{[t−1]} in its initial degree t. Thus, the study of HP OW LIN(r + 2, j, r)_{i} in its socle degree i = (j − 1)t corresponds to studying the related HP OINT S(s, a, n)_{i}, n = r − 1, a = i + 1 − j = (j − 1)(t − 1) in the initial degree i of the vanishing ideal (see [I1[§2.2]]). III. Related problems. A good non-computational proof of the case r = 8 of (3), would be a natural first goal. We answer the following question for s = r + 2, and (j, i) = (2, t), r = 2t in Corollary 8 below,

Question 5 For which values of (j, i) is

HP OW LIN(r + 2, j, r)_{i} − HF ROBERG(r + 2, j, r)_{i} > 0?  (10)

Note that R. Fröberg and J. Hollman [FH] have explored the Fröberg conjecture, that H(R/F (s, j)) = HF ROBERG(s, j, r) when j = 2, but we know of no analogous exploration of HP OW LIN(s, 2, r) in this exceptional case s = r + 2, except for that of [CEG] and some work of K. Chandler. This extreme case is a natural step in the larger program of understanding HP OW LIN(s, j, r).

Corollary 2.6, of [CEG] gives an inductive procedure for obtaining HP OINT S(s, A, n)_{i} where A = (a_{1}, … a_{s}), a_{1} ≥ … ≥ a_{s} is a sequence of vanishing orders at the s points. They treat either the case s ≤ n + 2 (the CI or almost complete intersection cases of HP OW LIN(s, j, r) when viewed dually), or degrees i satisfying i ≥ a_{1} + a_{n+3} − 1 that are high compared to the sequence of vanishing orders. Thus, by the dual correspondence mentioned above, their results bear on Question 5 in low degrees compared to the socle degree of R/L^{j}.

IV. Related results, and examples of Hilbert functions. IVA. Polynomial growth of length. Let P (J, r) = dim_{k}(R/(L_{1}^{j_{1}}, …, L_{s}^{j_{s}})), where J = (j_{1}, … j_{s}) is a sequence of positive integers, and s ≥ r. Note that when s = r, the complete intersection case, P (J) = j_{1} · . . . · j_{s}. Clare D’Cruz considered the question whether P (J, r) could be a multivariable polynomial, when s ≥ r, and L_{1}, …, L_{s} are generic linear elements of R, no two colinear, and showed

Proposition 6 (D’Cruz) When s = r + 1, and L is any fixed set of linear forms, no two colinear, then P (J, r) is never a multivariable polynomial.

Her proof uses a result of W. Brown [Br]. The negative result leaves open the question, whether P (s, j, r) = P (j, …, j) = HP OW LIN(s, j, r)(1) could be a polynomial in j.

IVB. Hilbert function information.

(s,j,r) HPOWLIN(s,j,r) HGEN(s,j,n), n=r-1
(6,2,4) (1, 4, 4)  (1, 4, 4)
(8,2,6) (1,6,13,8)  (1,6,13,8)
(10,2,8) (1,8,26,40,16)  (1,8,26,40,15)
(12,2,10) (1,10,43,100,121,32)  (1,10,43,100,121,22)
(14,2,12) (1,12,64,96,364,364,64)  (1,12,64,96,364,364,0)
(16,2,14) (1,14,89,336,820,1288,1093,128) (1,14,89,336,820,1288,1092,0)

HFROBERG(16,2,14)=(1,14,89,336,820,1288,1092,0)

IVC. A lower bound for HP OW LIN(r + 2, 2, r)

By a variant of the argument used in the case r odd, it is possible to show

Proposition 7 Let t ≥ 2 and r = 2t. Then HP OW LIN(r + 2, 2, r)_{t} ≥ r/2 + 1.

Indication of Proof. We use the geometric translation (IIBiii), and show dim_{k}I_{P} _{t}^{(t−1)} ≥ t + 1. The relevent degree-t form for the odd case r = 2t − 1, showing HP OW LIN(r + 1, 2, 2t − 1)_{t} ≥ 1 (see Equation (4) of Observation 1) is

F (2t − 1, C) = det ( \begin{matrix}x_{1} & c_{1}x_{1} & c_{1}^{2}x_{1} & … c_{1}^{t−1}x_{1} & c_{1} & c_{1}^{2} & … & c_{1}^{t−1} \\ x_{2} & c_{2}x_{2} & c_{2}^{2}x_{2} & … c_{2}^{t−1}x_{2} & c_{2} & c_{2}^{2} & … & c_{2}^{t−1} \\ … & & & … & & & … \\ x_{r} & c_{r}x_{r} & c_{r}^{2}x_{r} & … c_{r}^{t−1}x_{r} & c_{r} & c_{r}^{2} & … & c_{r}^{t−1} \end{matrix} ) . (7)

The form F (2t−1, C) is nonzero of degree-t, and vanishes to order t−1 at the point set P mentioned above – the coordinate points plus (1, 1, … 1) and (c_{1}, …, c_{2t−1}). Equivalently, F (2t − 1, C) annihilates (f, g)_{t} in the Macaulay duality on V_{t} (See Lemma 6.4 of [I2], based on an observation by M. Boij, but likely classical). For even r = 2t, one may define by restriction r such forms F_{1}(r, C), …, F_{r}(r, C), where F_{i}(r, C) = F (r − 1, C)|_{x_{1},...x̌_{i},...x_{r}}; the first t + 1 of these are linearly independent, since the i-th has a monomial cx_{1} · x_{2} ⋯ x̌_{i} ⋯ x_{t+1} not appearing in the others.

Corollary 8 . If t ≥ 4, and r = 2t then

HP OW LIN(r + 2, 2, r)_{t} − HF ROBERG(r + 2, 2, r)_{t} > 0.  (11)

Proof: For t = 4, 5 one can easily verify Equation (11) (see table on page 7). We will show that HF ROBERG(r + 2, 2, r)_{t} = 0 for all t ≥ 6. Consider

\frac{(1 − Z^{2})^{r+2}}{(1 − Z)^{r}} = (1 − Z^{2})^{2}(1 + Z)^{r}
= (1 − 2Z^{2} + Z^{4}) (∑_{i=0}^{r} \binom{r}{i}Z^{i})

The coefficient of Z^{t} in the above polynomial is

\binom{r}{t} − 2\binom{r}{t − 2} + \binom{r}{t − 4} = \frac{r!}{t!(t + 4)!} (−4(2t + 1)(t + 1)(t − 6)) .

It is easy to see that for t ≥ 6 the coefficient of Z^{t} cannot be positive hence that HF ROBERG(r+2, 2, r)_{t} = 0. Lemma 7 implies that HP OW LIN(r+2, 2, r)_{t} > 0.

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