2017/03/02 by А. А. Кытманов, А. С. Тихомиров, Kytmanov, Alexey +5
Computer Science · Mathematics · #14D20 #14E08 #14J60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1703.00710
openalex publication_date 2017/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the problem of rationality of an infinite series of components, the\nso-called Ein components, of the Gieseker-Maruyama moduli space M(e,n) of\nrank 2 stable vector bundles with the first Chern class e=0 or -1 and all\npossible values of the second Chern class n on the projective 3-space. The\ngeneralized null correlation bundles constituting open dense subsets of these\ncomponents are defined as cohomology bundles of monads whose members are direct\nsums of line bundles of degrees depending on nonnegative integers a,b,c,\nwhere b\≥ a and c>a+b. We show that, in the wide range when c>2a+b-e, \nb>a, (e,a)\≠(0,0), the Ein components are rational, and in the remaining\ncases they are at least stably rational. As a consequence, the union of the\nspaces M(e,n) over all n\≥1 contains an infinite series of rational\ncomponents for both e=0 and e=-1. Explicit constructions of rationality of\nEin components under the above conditions on e,a,b,c and, respectively, of\ntheir stable rationality in the remaining cases, are given. In the case of\nrationality, we construct universal families of generalized null correlation\nbundles over certain open subsets of Ein components showing that these subsets\nare fine moduli spaces. As a by-product of our construction, for c1=0 and\nn even, they provide, perhaps the first known, examples of fine moduli spaces\nnot satisfying the condition "n is odd", which is a usual sufficient\ncondition for fineness.\n