ACM curves in multiprojective spaces
type: Item
instance of: scholarly article
title: ACM curves in multiprojective spaces
zbMATH Open document ID: 1499.14088
author: E. Ballico
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2022-10-21T00:00:00Z
review text: A \textit{multiprojective space} is a product \(Y = \prod_{i=1}^k \mathbb P^{n_i}\) of projective spaces. The Picard group of \(Y\) consists of the line bundles \(\mathcal O_Y (d_1, \dots, d_k) = \otimes \pi_i^*(\mathcal O_{\mathbb P^{n_i}} (d_i))\) with \((d_1, \dots, d_k) \in \mathbb Z^k\). A closed subscheme \(T \subset Y\) is Arithmetically Cohen-Macaulay (ACM) if the restriction maps \(H^0(\mathcal O_Y (b_1, \dots, b_k)) \to H^0(\mathcal O_T (b_1, \dots, b_k))\) are surjective for all \((b_1, \dots, b_k) \in \mathbb N^k\). Extending the well known case \(k=1\), the author proves that if \(T \subset Y\) is ACM and \(\dim T = 1\), then \(T\) is connected with no embedded points; further, if \(\pi_i|_T\) is finite, then \(\pi_i|_T\) is an embedding, in particular \(\pi_i|_T\) is an isomorphism if \(n_i = 1\). The main results are these. If \(T \subset Y\) is an integral, non-degenerate ACM curve of multidegree \((a_1, \dots, a_k)\) and some \(n_i = 1\), then \(T\) is a smooth rational curve and each projection \(\pi_h|_T\) embeds \(T\) as a rational normal curve in \(\mathbb P^{n_h}\). If some \(n_i=2\), then \(T\) is isomorphic to a plane curve of degree \(a_i\). If \(Y = (\mathbb P^1)^k\), then \(T\) is ACM \(\iff T\) is connected and has multidegree \((1,\dots,1) \iff T\) is connected, has multidegree \((1, \dots, 1)\) with irreducible smooth rational components and at worst seminormal singularities and arithmetic genus \(p_a (T) = 0\). For example, in \(\mathbb P^1 \times \mathbb P^1\), take \(T\) to be the union of the coordinate axes. \{ Reviewer's remark: The author defines \(T \subset Y\) to be non-degenerate if (a) there is no multiprojective space \(T \subset Y^\prime \subset Y\) and says this is equivalent to (b) each projection \(\pi_i (T)\) is contained in no hyperplane in \(\mathbb P^{n_i}\). The diagonal inclusion \(\Delta: \mathbb P^n \hookrightarrow \mathbb P^n \times \mathbb P^n\) shows that these two are not equivalent. Looking at the paper, it …
reviewed by: Scott Nollet
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 7604916
zbMATH Keywords: multiprojective spaces
zbMATH Keywords: Hilbert function
zbMATH Keywords: ACM curve
zbMATH Keywords: arithmetically Cohen-Macaulay
MaRDI profile type: Publication
full work available at URL: s40574-022-00319-7
OpenAlex ID: W4210502216
cites work: The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves
cites work: Q5698935
cites work: On postulation of curves: Embeddings by complete linear series
cites work: Q4046174
cites work: On the Geometric Interpretation of Seminormality
cites work: Determinantal Equations for Curves of High Degree
cites work: Q4143433
cites work: Q3096060
cites work: Q4853308
DOI: 10.1007/S40574-022-00319-7
description: scientific article; zbMATH DE number 7604916