Asymptotic Hilbert Polynomial and a bound for Waldschmidt constants

In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasinska, Adv. Math. 2014. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.


Introduction
In the recent years the asymptotic invariants of ideals have aroused great interest and have been studied by many researchers, see for example [8], [9], [6], [7], [16], [15], [17] and others. One of these asymptotic invariants is the so called Waldschmidt constant of an ideal I ∈ K[P n ] denoted by α(I), see for example [3], [4], [2]. The constant is the limit of a sequence of quotients of the initial degrees of the m-th symbolic power of the ideal by m (see Definition 4). Computing this constant is a hard task in general, as it is difficult to compute the initial degree of a symbolic power of an ideal. For example, if I is the ideal of s points in P 2 in generic position, finding α(I) means finding the Seshadri constant of these points. The Seshadri constant is defined as the infimum of the quotients degC m1+...+ms , where C is a curve passing through P 1 , . . . , P s with multiplicities m 1 , . . . , m s . By the famous Nagata conjecture (see eg [5] or [1] and the references therein) we expect in this situation the equality α(I) = √ s, for s ≥ 10. Here we know that α(I) ≤ √ s, but in general we have no bounds on α(I) at all. The result of Esnault and Viehweg in [11] gives a lower bound of the Waldschmidt constant of an ideal of distinct points in P n .
In [5] the authors give an upper bound of α(I) in case I is an ideal of a sum of disjoint linear subspaces of P n , see Theorem 10. In the present paper we generalize this result and give an upper bound of α(I) for a wide class of ideals (namely, radical ideals with linearly bounded regularity of symbolic powers, see Preliminaries for the definitions). To find this bound we use aHP I (t), the so called asymptotic Hilbert polynomial of I, defined in [7]. The bound is given by the root of a suitable differential of the polynomial Λ I (t) := t n n! − aHP I (t). The main result of the present paper is the following theorem: Main Theorem. Let I be a radical homogeneous ideal in K[P n ] with linearly bounded regularity of symbolic powers. Assume that in the sequence depth I (m) there exists a constant subsequence of value n − c. Then Λ (c) The paper is organized as follows. In the second section we recall the necessary notions, in the third we prove the main result. The fourth section contains some interesting examples. In particular, in Example 8 we show that it is necessary to take the root of a derivative of the polynomial Λ I , not of the polynomial itself, as Λ I ( α(I)) > 0. Example 6 shows that we may get worse bounds on α(I) by computer-aided computations than by application of the Main Theorem.

Preliminaries
In the paper [7] the authors define the so called asymptotic Hilbert function and asymptotic Hilbert polynomial. Namely, let K be an algebraically closed field of characteristic zero, by K[P n ] = K[x 0 , . . . , x n ] we denote the homogeneous coordinate ring of the projective space P n . Let I be a homogeneous radical ideal in K[P n ], let I (m) denote its m-th symbolic power, defined as: where localizations are embedded in a field of fractions of K[P n ] ( [10]). By the Zariski-Nagata theorem, for a radical homogeneous ideal I in a polynomial ring over an algebraically closed field, the m-th symbolic power I (m) is equal to where m p denotes the maximal ideal of a point p, and V (I) denotes the set of zeroes of I. In characteristic zero, the symbolic power (of a radical ideal) can also be described as the set of polynomials which vanish to order m along V (I); this (compare [18]) can be written as: To define the asymptotic Hilbert polynomial, recall that the Hilbert function HF I of a homogeneous ideal I is defined as For t big enough the above function behaves as a polynomial, the Hilbert polynomial HP I of I. Let us define ideals with linearly bounded symbolic regularity (ie satisfying LBSR condition): Definition 1. Let I be a homogenous ideal in K(P n ). We say that I satisfies linearly bounded symbolic regularity, or is LBSR for short, if there exist constants a, b > 0 such that It is worth observing, that we do not know, so far, any example of a homogeneous ideal, which is not LBSR. The list of ideals which are proved to be LBSR may be found eg in [7]. Now we recall the definitions of the asymptotic Hilbert function and the asymptotic Hilbert polynomial of an ideal I.
m n in case that the limit exists.
In [7] it is shown that if I is a radical ideal then the limit exists.
m n in case that the limit exists.
In [7] it is shown that if I is a radical LBSR ideal then the limit exists. Recall the definition of the Waldschmidt constant of an ideal I: where α(J) is the least degree of a nonzero polynomial appearing in J (called the initial degree of J).
In [5] the authors defined a polynomial Λ n,r,s (t), namely let L be a sum of s disjoint linear subspaces of dimension r in P n (called a flat ; by a fat flat we denote such subspaces with multiplicities), as in [5]. Let I be the ideal of L. Define P n,r,s,m (t) := t + n n − HP I (m) (t).
Substitute t by mt into P n,r,m,s (t) and regard it as a polynomial in m (this is indeed a polynomial, see [5]). The leading term of this polynomial is denoted by Λ n,r,s (t).
In [7] it is shown that where I is the ideal of the fat flat.
In this paper we define Λ I (t) for any radical LBSR ideal as The main result of our paper gives an upper bound for α(I) in terms of the largest root of a suitable derivative of Λ I .

Main Result
The main result of our paper is the theorem below, giving an upper bound for α(I), where I is radical and satisfies LBSR condition. Thus, this theorem generalizes the Theorem 2.5 from [5], where the bound is proved for ideals of linear subspaces only. More comments on this generalization are in Remark 9.
In this case there is a subsequence of depth n.
We have to prove m n is less than or equal to zero for t = α(I).
Recall that Take If we prove that then we are done as Λ I is continuous.
So, it is enough to show that in our case for all nonnegative integers t ≥ α(I (m) ) − 1.
For this, observe that for any ideal J in K[P n ], we have that HF J = HF gin(J) , and HP J = HP gin(J) (where gin(J) is the initial ideal of J, with respect the degree reverse lexicographical order, of a generic coordinate change of J, cf the beginning of Section 4).
Ideal I is radical thus saturated. Since depth gin(I) = depth I = n, by Lemma 3.1 from [14] gin(I) involves all but one of the variables. Hence it is enough to prove the claim (1) for such ideals.  We will show by induction on n that ∆ is finite. For n = 1 the claim is obvious (since K = J is this case). So let n be arbitrary. By deg j (µ) we denote the degree of µ with respect to x j . For each j = 0, . . . , n − 1 let Observe that each d j is defined, as each variable appears as a factor in some generator. Let K j be the dehomogenization of K with respect to x j , let µ j be the dehomogenization of µ. Observe that the dehomogenization of each µ ∈ ∆ j belongs to the set ∆ defined for K j and µ j (the assumption that deg j (µ) ≥ d j plays a crucial role here). Hence, by inductive assumption (and since the degree of µ is bounded) each set ∆ j is finite. Observe also that Each of these sets is finite, hence ∆ is finite, as claimed. From finiteness of ∆ is follows that, for t big enough (it is enough to take t bigger than the maximum degree of a monomial in ∆), HF K (t) = HF J (t) + #∆. The same holds for Hilbert polynomials for all t. Observe that: Therefore, for t ≥ α(J), and as a consequence for t ≥ α(K) − 1, The last inequality follows from the fact that Since aHP I (t) is a polynomial by [7,Theorem 13], the derivative aHP ′ I (t) exists and is equal to the limit of the difference quotient. Moreover for any T ∈ R and for t ∈ [0, T ], aHP I is a limit of uniformly convergent polynomials with bounded degrees. Hence We proceed by induction on c.

Examples
In this section we present some interesting examples. There are two types of examples. The first type of examples concerns "crosses". By a "cross" we mean two intersecting lines. We show in Example 6, that our theorem gives better bound than those possible to compute with help of a computer.
The second type of examples are examples of some star configurations in P 4 . In particular, in Example 8 we show that it is necessary to take the root of a derivative of the polynomial Λ I , not of the polynomial itself. We also formulate a problem, which may be viewed as a generalization of Nagata and Nagata-type conjectures, see [5].
In the sequel we will need the notion of a limiting shape and some results from [6] and [7]. Let us start with recalling the notion of limiting shapes.
To define the limiting shape, consider first generic initial ideal gin(I) of I, as the initial ideal, with respect the degree reverse lexicographical order, of a generic coordinate change of I. Galligo [12] assures that for a homogeneous ideal I and a generic choice of coordinates, the initial ideal of I is fixed, hence the definition of gin(I) is correct.
In the next step consider the sequence of monomial ideals gin I (m) . The m-th symbolic power of a radical ideal I is saturated, hence by Green [13,Theorem 2.21] no minimal generator of gin I (m) contains the last variable x n . Therefore these monomial ideals can be naturally regarded as ideals in K[x 0 , . . . , x n−1 ]. The Newton polytope of a monomial ideal is defined as a convex hull of the set of exponents: P (J) := conv({α ∈ R n : x α ∈ J}).
The limiting shape of an ideal I as above is defined as (see Mayes [15]). Define Γ I as the closure of the complement of ∆(I) in R n ≥0 . Theorem 3 in [7] says that for radical LBSR ideal I and for T t = {(x 1 , . . . , x n ) : Now we present the first type of examples. First, take a "cross", ie two intersecting lines, in P 3 . A cross is a complete intersection of type (2, 1). From the results of Mayes [16], (Theorem 3.1) we have that for the ideal I of a cross gin( where T (m) is a triangle in R 2 with vertices (0, 0), (m, 0), (0, 2m). From this we have that the asymptotic limiting shape (see [16] or [6]) Thus we obtain that the asymptotic Hilbert polynomial of a cross in P 3 (4) aHP(t) = t − 1.
Indeed, by equation (3) we compute the volume of Γ I cut by the plane x + y + z = t, ie In the examples below we will also need the following, rather obvious fact: if Z I and Z J are two disjoint sets given as zero sets of radical LBSR ideals I and J respectively then (5) aHP I∩J = aHP I + aHP J .
The formula follows from the definitions of the Hilbert polynomial and the asymptotic Hilbert polynomial, from the exact sequence: and from the fact that for the ideals of disjoint sets R/I = (I + J)/I ≃ J/I ∩ J. Consider s generic crosses in P 3 with the ideal I s . Take the polynomial Λ s (see (4) and (5)): Denote the largest real root of Λ s by γ s . For s = 2, 3, 4 we have that γ 2 = 2.76873..., γ 3 = 3.60687..., γ 4 = 4.29021.... As in the same time α(I s ) = s, we have that α(I s ) ≤ s, which is less than the root of Λ s . In case s = 5 the situation is different: Example 6. Consider 5 generic crosses in P 3 with the ideal I 5 . Here Λ 5 (t) = t 3 6 − 5(t − 1) and α(I 5 ) = 5, moreover, using a computer we may check that for m = 2, . . . , 10 we still have α(I (m) 5 ) = 5m (and then the time of computations grows rapidly), but from our theorem we immediately know that α(I 5 ) ≤ γ 5 = 4.88447.... Remark 7. We may want to compute (with help of a computer) the expected initial degree eα m of I (m) (in the similar way as it was done for fat flats in [5]). This could be the way to bound α by finding the lowest possible term (or infimum) of eαm m . There are two problems with this method. The first is that the expected degree eα m may go down very slowly. For five crosses eα m = 5 up to m = 12 and eα13 13 = 64 13 = 4, 92307... > 4.88447.... The second problem is more important. We do not know if the expected initial degree is properly computed. The formula for a dimension of a system of forms of degree d vanishing along a given set with multiplicity m may be correct only for d big enough. We have an unpublished result that for a cross in P 3 the formula is correct for d ≥ 2m − 2.
Now we move to the second type of examples, concerning star configurations. We begin with giving an experimentally found formula for the asymptotic Hilbert polynomial of a star configurations given in P n by intersecting every c out of s generic hyperplanes. We will denote the ideal of such a configuration by I c,s,n . From Theorem 1.1 in [6] we know that Γ Ic,s,n = Γ Ic,s,c × R n−c , where Γ c,n,c is a simplex in R c with vertices s c , s−1 c−1 , . . . , s − (c − 1). Using equation (3) we see that to compute aHP Ic,s,n (t) it is enough to compute the volume of Γ Ic,s,n cut by the plane x 1 + . . . + x n = t. Denote The volume is the integral: By computing the integral for small values of n and c we found the formula: where the sum k denotes the sum of all monomials of degree k in variables a i . So far we are not able to prove the formula.
The next, important example shows that it is necessary to take an appropriate derivative of the polynomial Λ I . ). Remark 9. Theorem 2.5 in [5] gives a bound for the Waldschmidt constant of the ideal of the disjoint sum of linear subspaces (flats) in P n . Namely, it says that: Theorem 10. Let n, r, s be integers with n ≥ 2r + 1, r ≥ 0 and s ≥ 1. Let I be the ideal of s disjoint r-planes in P n . Then the polynomial Λ I (t) has a single real root bigger than or equal to 1. Denote this largest real root by γ I . Then α(I) ≤ γ I .
In particular the theorem holds for one linear subspace of codimension n − r. Observe, that it is in a sense accidental that in this case α(I) is bounded from above by the root of Λ I , as according to Theorem 5 we should take the largest root of the r-th derivative of the polynomial.
Next, we formulate a problem which may be viewed as a generalization of Nagata-type conjectures (see [5]). Note, that one may generalize the conjecture of Nagata asking if there exists a number N 0 (depending on an algebraic variety X), such that for s ≥ N 0 the Waldschmidt constant of the ideal of s generic points on X is maximal possible. The original Nagata conjecture says that N 0 = 10 for P 2 , and the maximal possible value of α is √ s, ie the largest root of the polynomial Λ for these points.
Problem 11. Let X be an algebraic variety. Take a radical ideal I in K[X]. Take the ideal where φ j , j = 1, . . . , s, is a generic change of coordinates. Then, for s big enough, the Waldschmidt constant of J is maximal possible, ie equal to the largest root of the suitable derivative of the polynomial Λ J (t) = t n n! − s · aHP I (t).