The moduli scheme \(M(0,2,4)\) over \(\mathbb{P}_ 3\)
type: Item
instance of: scholarly article
title: The moduli scheme \(M(0,2,4)\) over \(\mathbb{P}_ 3\)
zbMATH Open document ID: 0837.14008
DOI: 10.1007/BF02572323
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174652
review text: It is well-known that there is a coarse moduli scheme \(M(c_1, c_2, c_3)\) of semistable coherent sheaves of rank 2 on \(\mathbb{P}_3\) with Chern classes \(c_1, c_2, c_3\); although so far very few of these schemes have been studied in detail and not much is known about their structure in general. This work is devoted to the study of \(M(0,2,4)\), which is among the first nontrivial cases with extremal third Chern class, but still allowing explicit considerations. The open subset \(M_r\) of \(M(0,2,4)\) of \(\mu\)-stable reflexive sheaves was described by \textit{M.-C. Chang} [Trans. Am. Math. Soc. 284, 57-89 (1984; Zbl 0558.14015)] and \textit{C. Okonek} [J. Reine Angew. Math. 338, 183-194 (1983; Zbl 0491.14010)]. They proved that \(M_r\) is irreducible, rational and smooth of dimension 13. Moreover, they showed that any \({\mathcal F} \in M_r\) can be represented as a cokernel of a \(2 \times 4\)-matrix of linear forms, i.e. has a short resolution \[ 0 \to k^2 \otimes {\mathcal O} (-2) \to k^4 \otimes {\mathcal O} (-1) \to {\mathcal F} \to 0. \] It turned out that \(M_r\) is dense in the Maruyama scheme \(M(0,2,4)\) and that this compactification is the G.I.T.-quotient of the space of all semistable \(2 \times 4\)-matrices of linear forms under the natural action of \(\text{GL} (2) \times \text{GL} (4)\). Thus the study of \(M(0,2,4)\) is the study of \(2 \times 4\)-matrices of linear forms on a four-dimensional vector space. There are some remarkable and astonishing phenomena related to these matrices. In section 3 we succeeded in giving normal forms to these matrices in the different geometric situations related to their Fitting ideals of quadrics. This helps us to prove results on the subvarieties of the sheaves with specified geometrical data. -- Our purpose is to describe the most important subvarieties of \(M(0,2,4)\) and their sheaves. This classification is still far from revealing the structure of \(M(0,2,4)\) completely. In sections 1 and 2 we determine…
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 704919
zbMATH Keywords: moduli space \(M(0,2,4)\)
zbMATH Keywords: Maruyama scheme
author: Rosa M. Miró-Roig
author: Günther Trautmann
MaRDI profile type: Publication
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OpenAlex ID: W2030421864
description: scientific article; zbMATH DE number 704919