Regularity and symbolic defect of points on rational normal curves

In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective 3-space. We give an explicit formula for the value of Castelnuovo–Mumford regularity for their ordinary powers. Moreover, we compare the m-th symbolic and ordinary powers for such ideals in order to show whenever the m-th symbolic defect is non-zero.


Introduction
Studying Castelnuovo-Mumford regularity reg(I) of a homogeneous ideal I ⊆ K[x 0 , . . ., x n ] has a long story starting from the paper of Mumford [15], who introduced the concept of m-regularity for an ideal I, i.e. the number m for which all i-th syzygies of I are generated in degrees not greater than m + i, for all i.Bayer and Stillman in [1]went on with Mumford's ideas by showing an explicit criterion for m-regularity.They also proved an equality between reg(I) and the regularity of initial ideal of I with respect to the reverse lexicographic order in any characteristic of K.A connection between Castelnuovo-Mumford regularity and syzygies of given ideal I justifies why reg(I) can be viewed as a measure of complexity of I and also explain unflagging interests in this subject.
Swanson in [16] analyses r-th ordinary powers I r of homogeneous ideals I, showing that these powers can be expressed in terms of primary decomposition of I r .As an additional result, it has been proved that reg(I r ) is bounded above by some linear functions which depend on r.As a consequence, a new way of investigation of reg(I r ) has begun.In [9] Cutkosky, Herzog, and Trung, building upon papers of Swanson and the paper of Bertram, Ein and Lazarsfeld [2], introduced a new asymptotic invariant, the so-called asymptotic regularity areg(I) of a homogeneous ideal I. Later, the work on regularity of homogeneous ideals and their powers was significantly improved in [7,11], for instance for the case of Gorenstein and zero dimensional ideals.
One of the best known classes of curves in projective spaces P n are rational normal curves and they have been studied widely, see [4,6,5,8].Studying schemes of fat points lying on rational normal curves has its own long history.In [6] Catalisano and Gimigliano gave an algorithm for computing the Hilbert function for fat point schemes lying on a twisted cubic curve and they extended the work for rational normal curves in P n together with Ellia [5].At the same time, Conca in [8] described the Hilbert function and resolution of symbolic and ordinary powers of ideals of rational normal curves.
Our motivation for this work is computing the regularity of powers of ideals of points on two types of rational normal curves, conic and twisted cubic curve.The main results of this paper concerning the regularity of powers of such ideals are Theorems 3.4 and 4.5 which can be summed up as follows: Theorem.Let n ∈ {2, 3} and let C ⊂ P n be a rational normal curve.Denote by I D j the ideal defining a set of s general points on C. Let 0 ≤ j < n be such that s = nd − j, then The paper is organized as follows.In Section 2 we recall all needed definitions and prove basic facts that are used through the paper.The first non-trivial case of a rational normal curve is a conic in P 2 .We dedicate Section 3 to this case.It culminates with the proof of Theorem 3.4.Section 4 is devoted to the study of twisted cubic curves and the culmination of this section is Theorem 4.5.The last section is a small step towards understanding the structure of symbolic powers of ideals I D j .We prove that for all integers m ≥ 3 there is , and state a conjecture about the relation between symbolic and ordinary powers of ideals I D j .

Preliminary
Let S = K[x 0 , . . ., x n ] be the graded ring of polynomials over an algebraically closed filed K. Let Denote by I = I 2 (M ) be the ideal generated by the 2-minors of M (known as the Hankel matrix).It is known that the ideal I defines the rational normal curve (RNC for short) in P n , which we denoted by C, the Veronese embedding of (2.1) Recall that for any homogeneous ideal J the Hilbert function HF(S/J, t) of S/J, for t ∈ N ∪ {0}, is the dimension over K of degree t homogeneous part of S/J.Remark 2.1.For the ideal I = I 2 (M ) the Hilbert function of S/I is known to be HF(S/I, t) = n(t + 1) − (n − 1), for t ≥ 0.
Let J ⊂ S be any homogeneous ideal.We denote by β ij (J) the (i, j)-th Betti number of J, i.e. the dimension of Tor S i (J, K) in degree j.By definition, the Castelnuovo-Mumford regularity reg(J) of J is reg(J) = max {j − i : It is convenient to write β(J) and α(J) for the maximum and the minimum degree of the minimal set of generators of J, respectively.In general we have that reg(J) ≥ β(J) and reg(S/J) = reg(J) − 1.
Definition 2.3.Let J ⊂ S be a homogeneous ideal.Then the asymptotic regularity of J is the real number areg(J) = lim r→∞ reg(J r ) r .
At it was shown in [9, Theorem 1.1], we have always that areg(J) = β(J r ) r , since it is known that β(J r ) is linear function which depends on r for all r 0. Let D j ⊂ C be a set of nd − j general points on the rational normal curve C ⊂ P n for integers d ≥ 2 and 0 ≤ j ≤ n − 1. Denote by I D j the ideal defining the set D j .In the following we study the ideal I D j and the next lemma is an observation that we need in order to prove that the forms of order rd does not vanish in I r D j .Proof.On the one hand from Lemma 2.4 and the fact that β(I r D j ) ≤ reg(I r D j ), we have that rd ≤ reg(I r D j ).On the other hand since I D j is a zero-dimension ideal generated at most in degree d, therefore from [11,Corollary 7.9] we have that reg Proof.One can see that where ξ i is the i-th primitive root of unity for i = 1, . . ., d − 1.By (2.1) we have that distinct points, therefore the desired result follows.Moreover, we conclude that no two hyperplanes {x 0 − ξ α x n = 0} and {x 0 −ξ β x n = 0}, with α = β, intersect C at the same point for all α, β ∈ {1, 2, . . ., d− 1}.
In the following sections, we study the regularity of I r D j where D j lies on a conic in P 2 , or on a twisted cubic curve (TCC) in P 3 .Since we are considering these points in two separate sections, we agree to use the same notation of C for both, conic and TCC.

Regularity of points on a conic
This section is devoted to study the regularity of I r D j where D j ⊂ C ⊂ P 2 .By the definition of ideal I, we have that Lemma 3.1.Let D j be a set of 2d − j distinct points in P 2 lie on C for d ≥ 2 and j ∈ {0, 1}.Then its defining ideal can be represented as: Proof.We proceed as follows: • Let j = 0.By Lemma 2.6 one can see that , the desired result follows from Lemma 2.6.Proposition 3.2.Let D j be as in Lemma 3.1.Then, Proof.Let j = 0. Then the syzygy matrices of S/I D 0 are as follows Therefore, we have its minimal free resolution 0 − → S(−d − 2) and from that reg(S/I D 0 ) = d.Accordingly, reg( Similarly, for j = 1, we compute the syzygy matrices for S/I D 1 , Hence, We see that reg Proof.
. Directly from the definition of ordinary power . Hence, the first syzygy matrix of S/I r D 0 is It is a straightforward computation that the second syzygy matrix can be express in the following manner .

Regularity of points on a TCC
Let n = 3.In this section we study the reg(I r D j ), where D j is a set of points 3d − j lie on the twisted cubic curve C defined by the following ideal, Lemma 4.1.The ideal I D j defines the set D j of 3d−j distinct points on C for j = 0, 1, 2.
Proof.We divide the proof into three cases as the following.
Since the plane {x 2 − x 1 = 0} does not contain any tangent line to C, we conclude that {( One can see that the line { x 2 , x 1 } is not tangent to C, moreover by Lemma 2.6, we have that Proof.We are looking for minimal free resolutions of the form for any ideal I D j .Since for any j we know the generators of ideals I D j , we can write matrices A i explicitly.
For the sake of the completeness, denote by With some aids of any algebraic software program, such as Macaulay2 [14], we compute the syzygy matrices of S/I D j .In case of j = 0 we have , therefore the minimal free resolution is While for j = 1 there is For the last remaining case, j = 2, the matrices are the following Hence we can write, By a straightforward calculation from the definition of regularity, we get the desired assertion.
The minimal free resolution of I D 1 , calculated in the previous theorem, gives us immediately the following corollary.Proof.
).The r-th power of I D 0 is as the following D 0 ⊂ J, therefore we have the following exact sequence: Hence we have that reg(J) ≤ max reg J I r D 0 , reg(I r D 0 ) .
We know that the degree of J is 3rd, therefore either HF(S/J, rd) is 3rd − 1 or 3rd.By contradiction assume that HF(S/J, rd) = 3rd−1.Hence, the first difference of the Hilbert function of S/J is as follows, So, by [13,Proposition 5.2] follows that V (J) contains a subset of rd + 2 collinear points having multiplicities r.It is a contradiction with the fact that V (J) has only subsets of at most 2r collinear points.Therefore, We conclude that reg(J) = rd + 1.We know that HF(S/(J/I r D 0 ), t) = HF(S/I r D 0 , t) − HF(S/J, t), ∀t ≥ 0.
Since the set minimal generators of I r D 0 has only one form of degree β(I r D 0 ) = rd, we conclude that HF(S/I r D 0 , t) − HF(S/J, t) = c ∈ Z + , for all t ≥ rd.Therefore, the Hilbert function of S/(J/I r D 0 ) is partially as follows: This follows that reg J Remarks in P n .It is natural to ask about the regularity of the same type of ideals in higher projective spaces.However, simply calculations can show that the formula for reg(I r D j ), with r > 1, is much more complicated than for cases of P 2 and P 3 , and can not be easily described.Thus, we dedicate this section to be a leading step on further investigations in this subject, by proving the lemma which concerns reg(I D j ).
One can easily observe that the proof of the fact that ideals I D j indeed describes the set of nd − j distinct points can be mimic from the proof of Lemma 4.1.Also the next remark is similar to the result obtained in Proposition 4.2.

Symbolic defect
Comparing symbolic and ordinary powers of ideals of points in P N has became very popular in recent years.There are a few different concepts which are concerning "ideal containment problem".In this section we want to analyse one of them in the case of ideals I D j .Let us recall first the definition of symbolic power of ideal.Remark 5.4.There is one missing case of sdefect(I D 1 , 2) in the statement of Theorem 5.3.We expect that sdefect(I D 1 , 2) = 0, however we do not have a theoretical proof of this hypothesis.
Motivated by numerous tests and observations that we made, we want to finish this section with a conjecture that we was not able to prove, but we believe to be true.

Lemma 2 . 4 .
Let D j be a set of nd−j points on rational normal curve C.Then, β(I r D j ) = rβ(I D j ) = rd.Proof.The proof directly follows from [10, Exercise A2.21, d].More precisely, I D j is an ideal in the symmetric algebra S/I (the coordinate ring of C) generated at most in degree d.Proposition 2.5.Let D j be as in Lemma 2.4.If r ≥ 2 and d ≥ 2, then rd ≤ reg(I r D j ) ≤ reg(I D j ) + (r − 1)d.
therefore by Lemma 2.6 the desired result follows.This completes the proof.Proposition 4.2.Let D j be as in Lemma 4.1.Then,

I r D 0 is at most rd − 1 .Theorem 4 . 5 .Corollary 4 . 6 .
From Proposition 2.5, we know that reg(I r D 0 ) ≥ rd, hence, reg J Let D j be as in Lemma 4.1.If r ≥ 2 and d ≥ 2, then (1) reg(I r D 0 ) = rd + 1, (2) reg(I r D 1 ) = reg(I r D 2 ) = rd, Proof.The proof of (1) is a direct consequence of Propositions 2.5,4.2 and Lemma 4.4.The proof for j = 1 follows from Propositions 4.2,2.5 and [7, Proposition 1.12.6].The last remaining case for j = 2 similarly the result follows from Propositions 2.5 and 4.2.The proof is complete.For the ideals I D j defined in Lemma 4.1, we have that areg(I D j ) = lim r→∞ reg(I r D j ) r = d.

Definition 4 . 7 .
Let n ≥ 4 and 0 ≤ j ≤ n − 1.Let I D j be the ideal of a set nd − j points on C defined by the ideal I = I 2 (M ) as follows,

Remark 4 . 8 .
For ideals I D j defined as in Definition 4.7, one can compute the reg(I D j ) as in Proposition 4.2 by writing their free resolutions or directly by computing their Hilbert functions,reg(I D j ) = d + 1, if 0 ≤ j < n − 1 d, if j = n − 1.

Definition 5 . 1 .
Let I be a homogeneous ideal in a polynomial ring R. For m ≥ 1, the m-th symbolic power of I is the idealI (m) = R ∩   p∈Ass(I) (I m ) p   ,where the intersection is taken over all associated primes p of I. and proceed by induction on k ≥ 1 in order to show that ∈ I 2k D 0,2 , Q 1 f k 0,2 ∈ I 2k+1 D 0,2 .

Conjecture 5 . 5 .
Let D j be a set of 3d − j general points on a TCC, where 0 ≤ j ≤ 2. Then1) I (m) D j ⊆ I r D j ifand only if m ≥ r + 1 for any integer r ≥ 2, in the case j = 0, 2. 2) I (m) D 1 ⊆ I r D 1 if and only if m ≥ r + 1 for r ≥ 3, and moreover, I (m) D 1 ⊆ I 2 D 1 if and only if m ≥ 2.