# zbMATH — the first resource for mathematics

The moduli scheme $$M(0,2,4)$$ over $$\mathbb{P}_ 3$$. (English) Zbl 0837.14008
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 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.

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14N05 Projective techniques in algebraic geometry 14L24 Geometric invariant theory
##### Keywords:
moduli space $$M(0,2,4)$$; Maruyama scheme
Full Text:
##### References:
 [1] [B] Barth, W.: Some properties of stable rank 2 vector bundles on $$\mathbb{P}$$ n . Math. Ann.226, 125–150 (1977) · Zbl 0417.32013 · doi:10.1007/BF01360864 [2] [C] Chang, M.C.: Stable rank 2 reflexive sheaves on $$\mathbb{P}$$3 with smallc 2 and applications. Trans. Am. Math. Soc.284, 57–89 (1984) · Zbl 0558.14015 [3] [F] Fogarty, F.: Algebraic families on an algebraic surface. Am. J. Math.90, 511–521 (1968) · Zbl 0176.18401 · doi:10.2307/2373541 [4] [H1] Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254, 121–176 (1980) · Zbl 0437.14008 · doi:10.1007/BF01467074 [5] [H2] Hartshorne, R.: Algebraic Geometry. (Grad. Texts. Math., vol. 52) Berlin Heidelberg New York: Springer 1977 [6] [I] Iarrobino, A.: Compressed algebras and components of the punctual Hilbert scheme. In: Casas-Alvero, E. et al. (eds.) Algebraic geometry. (Lect. Notes Math., vol. 1124, pp. 146–165) Berlin Heidelberg New York: Springer 1984 [7] [LB] Le Barz, P.: Platitude et non-platitude de certains sous-schémas de $$\mathbb{P}$$ n . J. Reine Angew. Math.348, 116–134 (1984) · Zbl 0518.14003 [8] [LP] Le Potier, J.: Fibrés de Higgs et systèmes locaux. Seminaire Bourbaki, Exp. 737, 1990–1991 [9] [M1] Maruyama, M.: Moduli of stable sheaves. I. J. Math. Kyoto Univ.17, 91–126 (1977) · Zbl 0374.14002 [10] [M2] Maruyama, M.: Moduli of stable sheaves. II. J. Math. Kyoto Univ.18, 557–614 (1978) · Zbl 0395.14006 [11] [M-F] Mumford, D., Fogarty, J.: Geometric invariant theory, 2nd ed., Berlin Heidelberg New York: Springer 1982 · Zbl 0504.14008 [12] [N-T] Narasimhan, M.S., Trautmann, G.: Compactification ofM P3(0,2) and Poncelet pairs of conics. Pac. J. Math.145, 255–365 (1990) · Zbl 0753.14004 [13] [O] Okonek, C.: Moduli extremer reflexiver Garben auf $$\mathbb{P}$$ n . J. Reine Angew. Math.338, 183–194 (1989) · Zbl 0491.14010 [14] [OSS] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. (Prog. Math., vol. 3) Boston Basel Stuttgart: Birkhäuser 1980 · Zbl 0438.32016 [15] [T] Trautmann, G.: Poncelet curves and associated theta characteristics. Expos. Math.6, 29–64 (1988) · Zbl 0646.14025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.