On syzygies of divisors on rational normal scrolls
type: Item
instance of: scholarly article
title: On syzygies of divisors on rational normal scrolls
zbMATH Open document ID: 1310.14046
author: Euisung Park
published in: Mathematische Nachrichten
publication date: 2014-10-09T00:00:00Z
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 6353328
zbMATH Keywords: divisor of a rational normal scroll
zbMATH Keywords: minimal free resolution
zbMATH Keywords: arithmetically Cohen-Macaulay
MaRDI profile type: Publication
full work available at URL: mana.201300085
OpenAlex ID: W1519992916
cites work: The Geometry of Syzygies
cites work: Q3789692
cites work: Weil divisors on rational normal scrolls
cites work: Q3995221
cites work: Koszul cohomology and the geometry of projective varieties. II
cites work: Q3919817
cites work: CLASSIFICATION OF BETTI DIAGRAMS OF VARIETIES OF ALMOST MINIMAL DEGREE
cites work: Bounds on Castelnuovo-Mumford regularity for divisors on rational normal scrolls
cites work: Arithmetically Buchsbaum divisors on varieties of minimal degree
cites work: On graded Betti numbers and geometrical properties of projective varieties
cites work: On syzygies of linear sections
cites work: Syzygies of canonical curves and special linear series
DOI: 10.1002/MANA.201300085
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review text: Let \(R\) denote the homogeneous polynomial ring of projective space \(\mathbb P^r\) over an algebraically closed field \(K\). Let \(Y \subset \mathbb P^r\) be a variety of minimal degree, i.e. one for which \(\deg(Y) = \text{codim}(Y)+1\). Such varieties have been completely classified. Let \(X \subset Y\) be a divisor, thought of as a subvariety of \(\mathbb P^n\), and assume that \(X\) is non-degenerate. Let \(I_X\) be the homogenous ideal of \(X\) in \(R\). The main object of study in this paper is the minimal free resolution of \(I_X\). It is known that if \(X\) is arithmetically Cohen-Macaulay (ACM) then the graded Betti numbers of \(I_X\) depend only on the degree of \(X\). This paper describes the graded Betti numbers in the non-ACM case, which is much more delicate. The graded Betti numbers are invariant in the divisor class of \(X\), and the author shows that the converse is almost true (giving the only exception). He carefully computes different important invariants in terms of the divisor class.
reviewed by: Juan C. Migliore
description: scientific article; zbMATH DE number 6353328