Resurgence numbers of fiber products of projective schemes
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Resurgence numbers of fiber products of projective schemes Sankhaneel Bisui1 · Huy Tài Hà1 · A. V. Jayanthan2 · Abu Chackalamannil Thomas1 Received: 22 May 2020 / Accepted: 3 November 2020 © Universitat de Barcelona 2020
Abstract We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase. Keywords Resurgence number · Asymptotic resurgence · Points · Symbolic powers · Containments between powers Mathematics Subject Classification 13F20 · 14N05 · 13A02 · 13P10
1 Introduction Inspired by the well-celebrated result of Ein et al. [11], Hochster and Huneke [19], and driven by a series of conjectures and questions due to Harbourne and Huneke [16], containments between symbolic and ordinary powers of ideals have evolved to be a highly active research topic in recent years (cf. [1–6, 9, 10, 21–24]). The resurgence number (defined by Bocci and Harbourne [3]) and the asymptotic resurgence number (defined by Guardo et al. [14]) are measures of the non-containments between these powers of ideals. Specifically, for a nonzero proper ideal I in a polynomial ring, the resurgence number and the asymptotic resurgence number of I are given by
* Huy Tài Hà [email protected] Sankhaneel Bisui [email protected] A. V. Jayanthan [email protected] Abu Chackalamannil Thomas [email protected] 1
Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA
2
Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India
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𝜌(I) = sup
{
m | (m) | I ⊈ Ir r |
}
and 𝜌a (I) = sup
{
} m | (mt) ⊈ I rt , t ≫ 0 . |I r |
It is easy to see that 𝜌a (I) ≤ 𝜌(I) for any ideal I. However, a priori, it is not quite clear how different these invariants could be. In fact, if one replaces the ordinary power of I by its integral closure and defines { } { } m | (m) m | (mt) ⊈ I rt , t ≫ 0 , 𝜌(I) = sup | I ⊈ I r and 𝜌a (I) = sup |I r | r | then the main result of a recent work of DiPasquale et al. [7] shows that
𝜌(I) = 𝜌a (I) = 𝜌a (I). Moreover, since these invariants are hard to compute, there are very few examples where 𝜌(I) and 𝜌a (I) are known explicitly (cf. [8]). Our goal in this paper is to study the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we exhibit a pathological example of the difference between these two invariants, and provide a large family of ideals for which the asymptotic resurgence number could be computed explicitly. Our results show that while the asymptotic resurgence of a fiber product of projective schemes can be computed via that of given schemes, it is not necessarily the case for the ] , X = Proj A∕I , and resurgence number. To be more specific, let A = 𝕜[ℙN𝕜 ] , B = 𝕜[ℙM 𝕜 Y = Proj B∕J , for homogeneous ideals I ⊆ A and J ⊆
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