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Asymptotic resurgences for ideals of positive dimensional subschemes of projective space. (English) Zbl 1296.13017

Recent work of Ein-Lazarsfeld-Smith [L. Ein et al., Invent. Math. 144, No. 2, 241–252 (2001; Zbl 1076.13501)] and M. Hochster and C. Huneke [Invent. Math. 147, No. 2, 349–369 (2002; Zbl 1061.13005)] raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. B. Harbourne and the reviewer [Proc. Am. Math. Soc. 138, No. 4, 1175–1190 (2010; Zbl 1200.14018)] defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero dimensional subschemes of projective space.
In this paper the authors take the first steps toward extending this work to higher dimensional subschemes: they introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals of smooth subschemes, generalizing what was previously known. They apply these bounds to ideals of unions of general lines in \(\mathbb{P}^N\).
Given a homogeneous ideal \(I\) in the homogenous coordinate ring \(k[\mathbb{P}^N]\) of projective space over a field \(k\), the containment problem is to determine the set \(S_I\) of pairs \((r,m)\) for which the symbolic powers \(I^{(m)}\) is contained in the ordinary power \(I^r\). The results of Ein-Lazarsfeld-Smith and Hochster-Huneke show that \(I^{(m)}\subset I^r\) whenever \(m\geq rN\) and hence \(\{ (r,m)\, :\, m\geq rN\} \subseteq S_I\). Even though there are many results on this topic, there are few cases for which \(S_I\) is known completely. Harbourne and the reviewer introduced the resurgence \(\rho(I)\) of an ideal \(I\): \[ \rho(I)=\sup \{m/r\, : \, I^{(m)}\not\subset I^r\}. \] The resurgence is a way if characterizing \(S_I\) numerically, but the resurgence itself has ben determined only in very special cases, in particular for ideals of zero-dimensional schemes in \(\mathbb{P}^N\).
In order to extend these results to higher dimensional subschemes, the authors introduce asymptotic refinements \(\rho_a(I)\) and \(\rho_a'(I)\) of the resurgence better adapted to studying ideal \(I=I(Z)\) defining higher dimensional subschemes \(Z\) of projective space \(\mathbb{P}^N\): \[ \rho_a(I)=\sup \{m/r\, :\, I^{(mt)}\not\subset I^{rt}\, \forall t\gg 0\}. \]
\[ \rho_a'(I)=\lim_{t\to \infty}\sup \rho(I,t) \] where \(\rho(I,t)=\sup \{m/r\, :\, I^{(m)}\not\subset I^{r}\, m\geq t, \, r\geq t\}\).
The main result of the paper gives upper and lower bounds on these asymptotic resurgences in terms of three numerical characters of \(I\) (viz., the least degree \(\alpha(I)\) and largest degree \(\omega(I)\) in a minimal homogeneous set of generators of \(I\), and the asymptotic invariant \(\gamma(I) = \lim_{m\to \infty} \alpha(I^{(m)})/m\)).
Theorem. Consider a homogeneous ideal \((0)\not= I\subset k[\mathbb{P}^N]\). Let \(h= \min(N, h_I )\) where \(h_I\) is the maximum of the heights of the associated primes of \(I\).
(1) We have \(1 \leq \alpha(I)/\gamma(I)\leq \rho_a(I) \leq \rho_a'(I) \leq \rho(I) \leq h\).
(2) If \(I\) is the ideal of a (non-empty) smooth subscheme of \(\mathbb{P}^N\), then \(\rho_a(I) \leq \omega(I)/\gamma(I) \leq \mathrm{reg}(I)/\gamma(I)\).
(3) If for some positive integer \(c\) we have \(I^{(cm)} = (I^{(c)})^m\) for all \(m \geq 1\), and if \(I^{(c)} \subseteq I^b\) for some positive integer \(b\), then \(\rho_a'(I)\leq c/b\).
The authors show an application of the previous theorem in interesting cases.
Corollary. Let \(I\) be the ideal of \(s\) general lines in \(\mathbb{P}^N\) for \(N \geq 3\), where \(s = {t+N \choose N}/(t + 1)\) for any integer \(t \geq 0\) such that \(s\) is an integer. Then \(\rho_a(I)=(t+1)/\gamma(I)\).
Theorem. Let \(I\) be the ideal of \(s\) general lines in \(\mathbb{P}^N\) for \(N \geq 2\), and \(s \leq (N + 1)/2\). Then \(\rho(I)=\rho_a'(I)=\rho_a(I)=\max(1,2s-1)\). Moreover, if \(2s<N+1\), then \(\gamma(I)=1\), while if \(2s=N+1\), then \(\gamma(I) = (N+1)/(N-1)\).
There is also the question of what we might expect \(\gamma(I)\) to be for the ideal \(I\) of \(s\) generic lines in \(\mathbb{P}^N\). The analogous question for ideals of generic points in \(\mathbb{P}^N\) is also still open in general but there are conjectures for what the answer should be. Nagata posed a still open conjecture for ideals of points in \(\mathbb{P}^2\), in connection with his solution of Hilbert’s 14th problem. In the terminology of the paper we can paraphrase it as follows.
Nagata’s Conjecture. Let \(I\subset k[\mathbb{P}^2]\) be the ideal of \(s \geq 10\) generic points of \(\mathbb{P}^2\). Then \(\gamma(I) = s^{\frac{1}{2}}\).
A generalization has been given separately by Iarrobino and Evain for generic points in \(\mathbb{P}^N\), which we paraphrase in the following way:
Conjecture. Let \(I\subset k[\mathbb{P}^N]\) be the ideal of \(s\geq 0\) generic points of \(\mathbb{P}^N\). Then \(\gamma(I) = s^{\frac{1}{N}}\).
It is of interest to see what we get if we apply the same reasoning to the problem of determining \(\gamma(I)\) for ideals \(I\) of generic lines in \(\mathbb{P}^N\). After some numerical results for computing \(\gamma\) on this setting, the authors end with the following conjecture
Conjecture. There is an integer \(q\) such that, for the ideal \(I\) of \(s \geq q\) sufficiently general lines (say \(s\) generic lines) in \(\mathbb{P}^3\), \(\gamma(I)\) is equal to the largest real root \(\tau = g\) of \(\tau^3 - 3s\tau + 2s\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
14C20 Divisors, linear systems, invertible sheaves

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CoCoA; Macaulay2
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References:

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