Demailly's Conjecture and the containment problem
Introduction
Let be a field, let be an integer, let be the homogeneous coordinate ring of , and let be its maximal homogeneous ideal. For a homogeneous ideal , let denote the least degree of a homogeneous polynomial in I, and let 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 with a prescribed order can be, Chudnovsky [13] made the following conjecture.
Conjecture 1.1 Chudnovsky Suppose that is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points . Then, for all ,
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 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 is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points and let be any integer. Then, for all ,
Demailly's Conjecture for 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 , and , for each there exists an open dense subset of the Hilbert scheme of s points in such that Demailly's bound (D) for holds for . As a consequence, Demailly's Conjecture holds for all . Chang and Jow [14] further proved that if , for some , then one can take to be the same for all , i.e., Demailly's Conjecture holds for a general set of points.
In this paper, we establish Demailly's Conjecture for a general set of sufficiently many points. More precisely, we show that given and , there exists an open dense subset U of the Hilbert scheme of s points in such that Demailly's bound (D) holds for and all .
Theorem 2.9 Suppose that is algebraically closed (of arbitrary characteristic) and . For a fixed integer , let I be the defining ideal of a general set of points in . For all , we have
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 of big height h we have To achieve the Stable Harbourne–Huneke Containment, we showed that one particular containment , for some value , would lead to the stable containment for . 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: 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 , for some value , would imply 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 . 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 . Our results are stated as follows.
Theorem 3.6 and Remark 3.9 Let be a field. Let I be the defining ideal of a codimension h star configuration in , for . For any , we have Let be the ideal of t-minors of a matrix X of indeterminates, and let h denote its height in . For all , we have
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 we have
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 matrix of indeterminates and let be the corresponding polynomial ring. For , let be the ideal in R generated by the t-minors of X; that is, is generated by the determinants of all submatrices of X. It is a well-known fact that is a prime and Cohen-Macaulay ideal of height .
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 be a collection of homogeneous polynomials in R and let be an integer. Suppose that any elements in form a complete intersection. The defining ideal of the codimension h star configuration given by is defined to be
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 are generated by “monomials” in the elements of . A similar description for symbolic powers of determinantal ideals 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 , the Waldschmidt constant of I is defined to be It is known (cf. [6, Lemma 2.3.1]) that the Waldschmidt constant of an ideal exists and Thus, Demailly's Conjecture can be equivalently stated as follows.
Conjecture 2.1 Demailly Let be an algebraically closed field of characteristic 0. Let 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 , or a homogeneous radical ideal of big height h in a polynomial ring R
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