The moduli scheme \(M(0,2,4)\) over \(\mathbb{P}_ 3\).

*(English)*Zbl 0837.14008It 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 M.-C. Chang [Trans. Am. Math. Soc. 284, 57-89 (1984; Zbl 0558.14015)] and 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 the subvarieties \(S_0, S_1, S_2\) of non-reflexive sheaves of \(M (0,2,4)\) and prove theorem I.

The reflexive sheaves \({\mathcal F}\) in \(M(0,2,4)\) are singular at a 0- dimensional subscheme \(\mathbb{Z} ({\mathcal F}) \subset \mathbb{P}_3\) of length 4. They can be classified by the multiplicities of the points of \(\mathbb{Z} ({\mathcal F})\). This leads to the closures \(D_\nu\) of the subvarieties of sheaves with a \(\nu\)-fold point in \(\mathbb{Z} ({\mathcal F})\). These are studied in section 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 the subvarieties \(S_0, S_1, S_2\) of non-reflexive sheaves of \(M (0,2,4)\) and prove theorem I.

The reflexive sheaves \({\mathcal F}\) in \(M(0,2,4)\) are singular at a 0- dimensional subscheme \(\mathbb{Z} ({\mathcal F}) \subset \mathbb{P}_3\) of length 4. They can be classified by the multiplicities of the points of \(\mathbb{Z} ({\mathcal F})\). This leads to the closures \(D_\nu\) of the subvarieties of sheaves with a \(\nu\)-fold point in \(\mathbb{Z} ({\mathcal F})\). These are studied in section 4.

##### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14N05 | Projective techniques in algebraic geometry |

14L24 | Geometric invariant theory |

##### References:

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