Demailly's Conjecture and the containment problem

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Abstract

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of this containment holds for generic determinantal ideals and defining ideals of star configurations.

Introduction

Let k be a field, let NN be an integer, let R=k[PkN] be the homogeneous coordinate ring of PkN, and let m be its maximal homogeneous ideal. For a homogeneous ideal IR, let α(I) denote the least degree of a homogeneous polynomial in I, and letI(n):=pAss(R/I)InRpR denote its n-th symbolic power. In studying the fundamental question of what the least degree of a homogeneous polynomial vanishing at a given set of points in PkN with a prescribed order can be, Chudnovsky [13] made the following conjecture.

Conjecture 1.1 Chudnovsky

Suppose that k is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points XPkN. Then, for all n1,α(I(n))nα(I)+N1N.

Chudnovsky's Conjecture has been investigated extensively, for example in [25], [6], [30], [27], [22], [21], [26], [5]. In particular, the conjecture was proved for a very general set of points [26] ([21] also proved the conjecture in this case but for at least 2N points) and for a general set of sufficiently many points [5]. The conjecture was also generalized by Demailly [16] to the following statement.

Conjecture 1.2 Demailly

Suppose that k is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points XPkN and let mN be any integer. Then, for all n1,α(I(n))nα(I(m))+N1m+N1.

Demailly's Conjecture for N=2 was proved by Esnault and Viehweg [25]. Recent work of Malara, Szemberg and Szpond [39], extended by Chang and Jow [14], showed that for a fixed integer m, Demailly's Conjecture holds for a very general set of sufficiently many points. Specifically, it was shown that, given N3, mN and s(m+1)N, for each n1 there exists an open dense subset Un of the Hilbert scheme of s points in PkN such that Demailly's bound (D) for α(I(n)) holds for XUn. As a consequence, Demailly's Conjecture holds for all Xn=1Un. Chang and Jow [14] further proved that if s=kN, for some kN, then one can take Un to be the same for all n1, i.e., Demailly's Conjecture holds for a general set of kN points.

In this paper, we establish Demailly's Conjecture for a general set of sufficiently many points. More precisely, we show that given N3,mN and s(2m+3)N, there exists an open dense subset U of the Hilbert scheme of s points in PkN such that Demailly's bound (D) holds for XU and all n1.

Theorem 2.9

Suppose that k is algebraically closed (of arbitrary characteristic) and N3. For a fixed integer m1, let I be the defining ideal of a general set of s(2m+3)N points in PkN. For all n1, we haveα(I(n))nα(I(m))+N1m+N1.

To prove Theorem 2.9, we use a similar method to the one we used in our previous work [5], where we proved Chudnovsky's Conjecture for a general set of sufficiently many points. This is not, however, a routine generalization. In [5], Chudnovsky's bound (C) was obtained via the (Stable) Harbourne–Huneke Containment, which states that for a homogeneous radical ideal IR of big height h we haveI(hr)mr(h1)Ir for r0. To achieve the Stable Harbourne–Huneke Containment, we showed that one particular containment I(hch)mc(h1)Ic, for some value cN, would lead to the stable containment I(hrh)mr(h1)Ir for r0. In a similar manner, Demailly's bound (D) would follow as a consequence of the following more general version of the (Stable) Harbourne–Huneke Containment:I(r(m+h1))mr(h1)(I(m))r for r0. Unfortunately, this is where the generalization of the arguments in [5] breaks down. We cannot prove that one such containment would lead to the stable containment. To overcome this obstacle, we show that the stronger containment I(c(m+h1)h+1)mc(h1)(I(m))c, for some value cN, would imply I(r(m+h1))mr(h1)(I(m))r for infinitely many values of r, and this turns out to be enough to obtain Demailly's bound.

It is an open problem whether, for a homogeneous radical ideal I, the general version of the Stable Harbourne–Huneke Containment stated in (HH) holds; this problem is open even in the case where I defines a set of points in PkN. In the second half of the paper, we investigate the general containment problem. We show that the containment holds for generic determinantal ideals and the defining ideals of star configurations in PkN. Our results are stated as follows.

Theorem 3.6 and Remark 3.9

Let k be a field.

  • (1)

    Let I be the defining ideal of a codimension h star configuration in PkN, for hN. For any m,r,c1, we haveI(r(m+h1)h+c)m(r1)(h1)+c1(I(m))r.

  • (2)

    Let I=It(X) be the ideal of t-minors of a matrix X of indeterminates, and let h denote its height in k[X]. For all m,r,c1, we haveI(r(m+h1)h+c)m(r1)(h1)+c1(I(m))r.

In particular, if I is the defining ideal of a star configuration or a generic determinantal ideal, then I satisfies a Demailly-like bound, i.e., for all n1 we haveα(I(n))nα(I(m))+h1m+h1.

Determinantal ideals are classical objects in both commutative algebra and algebraic geometry that have been studied extensively. The list of references is too large to be exhausted here; we refer the interested reader to [7] and references therein. In this paper, we are particularly interested in generic determinantal ideals. Specifically, for a fixed pair of integers p and q, let X be a p×q matrix of indeterminates and let R=k[X] be the corresponding polynomial ring. For tmin{p,q}, let It(X) be the ideal in R generated by the t-minors of X; that is, It(X) is generated by the determinants of all t×t submatrices of X. It is a well-known fact that It(X) is a prime and Cohen-Macaulay ideal of height h=(pt+1)(qt+1).

Star configurations have also been much studied in the literature with various applications [15], [10], [11], [46], [42], [47], [1], [4], [9]. They often provide good examples and a starting point in investigating algebraic invariants and properties of points in projective spaces; for instance, the minimal free resolution (cf. [2], [43]), weak Lefschetz property (cf. [44], [2], [37]), and symbolic powers and containment of powers (cf. [27], [31], [45], [38]).

We shall use the most general definition of a star configuration given in [38]. Let F={F1,,Fn} be a collection of homogeneous polynomials in R and let h<min{n,N} be an integer. Suppose that any (h+1) elements in F form a complete intersection. The defining ideal of the codimension h star configuration given by F is defined to beIh,F=1i1<<ihn(Fi1,,Fih).

To prove Theorem 3.6, Theorem 3.8, we use arguments similar to those in [8], where the containment has been proved for squarefree monomial ideals. Note that, by a recent result of Mantero [38], it is known that symbolic powers of the defining ideal of a star configuration Ih,F are generated by “monomials” in the elements of F. A similar description for symbolic powers of determinantal ideals It(X) was given by DeConcini, Eisenbud, and Procesi [17].

Acknowledgments

The second author thanks Jack Jeffries for helpful discussions. The second author is supported by the National Science Foundation, grant DMS-2001445. The third author is partially supported by Louisiana Board of Regents, grant LEQSF(2017-19)-ENH-TR-25. The authors thank the anonymous referee for a careful read and many useful suggestions.

Section snippets

Demailly's Conjecture for general points

In this section, we establish Demailly's Conjecture for a general set of sufficiently many points. Recall first that for a homogeneous ideal IR, the Waldschmidt constant of I is defined to beαˆ(I)=limnα(I(n))n. It is known (cf. [6, Lemma 2.3.1]) that the Waldschmidt constant of an ideal exists andαˆ(I)=infnNα(I(n))n. Thus, Demailly's Conjecture can be equivalently stated as follows.

Conjecture 2.1 Demailly

Let k be an algebraically closed field of characteristic 0. Let Ik[PkN] be the defining ideal of a set of

Harbourne–Huneke Containment beyond points

In this section, we investigate a general containment between symbolic and ordinary powers of radical ideals, and show that this containment holds for generic determinantal ideals and the defining ideals of star configurations. Specifically, we are interested in the following general version of the Harbourne–Huneke Containment for radical ideals.

Question 3.1

Let I be either a radical ideal of big height h in a regular local ring (R,m), or a homogeneous radical ideal of big height h in a polynomial ring R

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