The Effect of Points Fattening on del Pezzo Surfaces
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The Effect of Points Fattening on del Pezzo Surfaces 1 Magdalena Lampa-Baczynska ´
Received: 12 February 2020 / Revised: 2 August 2020 / Accepted: 1 September 2020 © The Author(s) 2020
Abstract In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces Sr arising by blowing-up of P2 at r general points, with r ∈ {1, . . . , 8}. Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces. Keywords Initial degree · Initial sequence · Blow-up · Alpha problem · Chudnovsky-type results Mathematics Subject Classification 52C30 · 14N20 · 05B30
1 Introduction In this paper, we follow the approach to fat point schemes initiated by Bocci and Chiantini in [3]. The initial degree α(I ) of a homogeneous ideal I ⊂ C[Pn ] is the least degree t such that the homogeneous component It in degree t is nonzero. Although this notion was known since 1981 (see [4]), Bocci and Chiantini used this invariant for the first time in order to study fat point subschemes in the projective plane. This definition can be extended to symbolic powers I (m) of I ; namely, α(I (m) ) is the least degree t such that the homogeneous component (I (m) )t in degree t is nonzero. Let Z ⊂ P2 (C) be a set of points and I be its radical ideal. By Nagata–Zariski theorem ([7], Theorem 3.14), the ideal of scheme m Z is the mth symbolic power of I . Bocci and Chiantini proved, among others, that sets of points Z in P2 (C) such that α(I (2) ) − α(I ) = 1,
Communicated by Rosihan M. Ali.
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Magdalena Lampa-Baczy´nska [email protected] Institute of Mathematics, Pedagogical University of Cracow, Cracow, Poland
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M. Lampa-Baczynska ´
are either contained in a single line or form the so-called star configuration. Results of Bocci and Chiantini have been generalized in [6] by Dumnicki, Szemberg and Tutaj-Gasi´nska. They were studying configurations of points in P2 (C) with α(I (m+1) ) − α(I (m) ) = 1 for some m ≥ 2 and obtained their full characterization (see [6], Theorem 3.4). These considerations were extended for another types of spaces. Except for spaces Pn , the problem of points fattening was considered among others by me in [1] for the space P1 × P1 and by Di Rocco, Lundman and Szemberg in [5] for Hirzebruch surfaces (with appropriately modified definition of the initial degree). The aim of this paper is to make similar classification with respect to points fattening on del Pezzo surfaces. In this paper, a del Pezzo surface is a smooth complex surface X with the ample anticanonical bundle −K X . In fact, considerations on points fattening effect were initiated on del Pezzo surface P2 (C) and this path of research was continued to another one, namely P1 × P1 . Over C, there are exactly ten del Pezzo surfaces: P2 , P1 ×P1 and eight surfaces Sr arising by blowing-up of P2 in r general points, w
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