Finding Ein components in the moduli spaces of stable rank 2 bundles on the projective 3-space (Q530287)

Let \(k\) be an algebraically closed field of characteristic \(0\). If \(c > b \geq a \geq 0\) are integers, for any set of homogeneous polynomials of degrees \(c-b, c-a, c+a\), respectively \(c+b\) in \(k[x_0,x_1,x_2,x_3]\), without common zeros, one has a monad \[ 0 \to {\mathcal O}(-c) \to {\mathcal O}(-b)\oplus {\mathcal O}(-a) \oplus {\mathcal O}(a)\oplus {\mathcal O}(b)\to {\mathcal O}(c)\to 0, \] where \({\mathcal O}:={\mathcal O}_{{\mathbb P}^3}\). The homology of this monad is a vector bundle of rank \(2\), with \(c_1=0\) and \(c_2=c^2-a^2-b^2\). Such bundles are the so-called Ein bundles. \textit{L. Ein} showed [Nagoya Math. J. 111, 13--24 (1988; Zbl 0663.14012)] that these bundles are stable iff \(c > a+b\) and in this case the moduli space \(M_{}(0,c^2-a^2-b^2)\) has an irreducible component \(N(a,b,c)\) in which Ein bundles as above are dense. The paper under review has the goal to find the Ein components and proposes a conjecture on the growth rate of the number of Ein components in terms of \(c_2\). The main method of the study is the use of spectra of bundles and their properties.