Resolution of general stable bundles of rang 2 over \(\mathbb P^3\) and Chern classes \(c_1 =-1,c_2 =2p\geq 6\). I
For \(c_1\in \{0,-1\}\), \(c_2\) even if \(c_1=-1\), let \(M(c_1,c_2)\) be the moduli space of rank \(2\) stable vector bundles on \(\mathbb {P}^3\) with Chern classes \(c_1,c_2\). At least if \(c_2\geq 6\) \(M(c_1,c_2)\) has an irreducible component whose general element \(F\) has natural cohomology [\textit{R. Hartshorne} and \textit{A. Hirschowitz}, Ann. SC. Éc. Supér. (4), 365--390 (1982; Zbl 0509.14015)]. In [\textit{A. Rahavandrainy}, C. R. Acad. Sci. Paris, Sér. I, Math. 325, No. 2, 189--192 (1997; Zbl 0918.14009)], the author computed the minimal free resolution of \(F\) if \(c_1=0\). Here he computes it and shows that it is the expected one (no line bundle appearing in two different places) when \(c_1=-1\) and \(c_2 > (v+2)(2v^2+3v-1)/(6v+7)\), where \(v\) is the minimal integer such that \(h^0(F(v))>0\). The proof uses the vector bundle Horace method as in the quoted papers and in \textit{M. Idà} [J. Reine Angew. Math. 403, 67--153 (1990; Zbl 0681.14032)].
stable vector bundle on \(\mathbb {P}^3\)
scientific article; zbMATH DE number 5799090