On moduli of stable 2-bundles with small Chern classes on \(Q_ 3\). Appendix by Nicolae Manolache: The curves \(Y\) of degree 6 with \(\omega_ Y= {\mathcal O}_ Y(-1)\) on a smooth quadric \(Q_ 3\)
Let \(M(c_1, c_2)\) be the moduli space of stable holomorphic vector bundles of rank two and Chern classes \(c_1\) and \(c_2\) over the smooth quadric \(Q_3\). It is known that for \(c_1 = - 1\), \(c_2 = 1\) this space consists exactly of the so-called spinor bundle (or more precisely its dual), while for \(c_1 = 0\) this moduli is empty for odd values of \(c_2\). In the paper under review, the authors study the moduli space of normalized (i.e. \(c_1 = 0\) or 1) such vector bundles for small values of \(c_2\), more precisely, for \((c_1, c_2) = (0,2)\), \((- 1,2), (- 1,3)\), \((0, 4)\). They focus their attention on \(M(0,2)\), which they study from different points of view. More precisely, they show that \(M(0,2)\) is isomorphic to \(\mathbb{P}^0\) minus a quartic hypersurface that they explicitly determine when identifying \(\mathbb{P}^9\) with the space of all \(5 \times 5\) skew-symmetric matrices or with the space of all quadrics in \(\mathbb{P}^3\). They also describe in three different ways \(M(0,2)\) as a nontrivial fibration over \(\mathbb{P}^4 \backslash Q_3\) with fibers \(P^5 \backslash Q_4\). For the other moduli spaces they prove that \(M (- 1,2)\) is a locally trivial fibration over \(Q_4 \backslash Q_3\) with fibers \(\mathbb{P}^2 \backslash Q_1\) and that \(M (- 1,3)\) and \(M (0,4)\) are unirational and reduced of respective dimensions 12 and 21. In the appendix, \textit{N. Manolache} classifies all curves (possibly reducible or non-reduced) \(Y \subset Q_3\) of degree 6 whose canonical line bundle is \(\omega_Y = {\mathcal O}_Y (-1)\). He obtains exactly nine families. This classification is used in the main paper for the study of \(M(0,4)\).
moduli space of normalized vector bundles
scientific article; zbMATH DE number 711497
scientific article