Math. Z. 184, 407-415 (1983) Mathematische Zeitschrift 9 Springer-Verlag 1983 Stable Rank 2 Bundles on ]p3 with r --0, C 2 =4, and g= 1 Mei-Chu Chang Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA The existence of the moduli of stable rank 2 vector bundles on IP 3 has been established by Maruyama [11]. So far, the moduli spaces have been described in detail only in a few cases: q=0, c2=1 (Barth [1], Wever [12]), c 1 =0, c2=2 (Hartshorne [7]), c 1 =- 1, c 2 =2 (Hartshorne-Sols [10]), c 1 =0, c2=3 (Ellings- rud-Stromme [6]), c1=0, c2=4, e=0 (Barth [2]). In this paper we do the case C1=0 , C2=4 , Cr In Sect. 1 we show that a general such bundle E is the cohomology bundle of a self-dual monad 0-*(9(-2)--.4(9~(9(2)--*0. In particular, the moduli space M(0, 4)~= 1 as a set is irreducible and rational of dimension 29. In section 2 we study the jumping lines of E. Our main technique is to use the reduction step introduced by Hartshorne [8,9.1] and dimension counting, namely, the fact [8, 3.4.1] that the dimension of any irreducible component of the moduli M(0, c2) is at least 8c 2 - 3. Acknowledgement. I would like to thank my thesis advisor Robin Hartshorne for some crucial ideas in this paper. I am also greatly indebted to the referee for a number of suggestions which have improved this paper. Indeed, among other things the present form of (1.5) was suggested by him. w Proposition 1.1. Let E be a semistable rank 2 reflexive sheaf on ~p3 with Chern classes (0, 1, c3), then c3=0 or 2. If c3=0, E is stable. If e3=2, E is only properly semistable. Proof Theorem 8.2 in [8] implies the first statement and H2(E)=0. Riemann- Roch gives z(E)= 89 3. Hence if c3--2 , then h~ and E is not stable. Con- versely, if E is properly semistable, then H~ 0, and we have the extension O-~ C --~ E--~ I L--* O. (1) So, c3=2P~-2+4d=2.