On the stability of the restricted bundle of space curves contained in a quartic surface
The author shows that, for \((d+13)^2/9\leq g<(d-1)^2/8\), there exist smooth connected curves \(Z\) in \(\mathbb{P}^3\), of degree \(d\) and genus \(g\), such that the restriction of the tangent bundle of \(\mathbb{P}^3\) to \(Z\) is stable. The main ingredients in his proof are a definition of stability for vector bundles on reduced but not necessarily irreducible curves due to \textit{G. Hein} and \textit{H. Kurke} [in: Proc. Hirzeburch 65 Conf. Algebraic Geometry, Ramat Gan 1993, Isr. Math. Conf. Proc. 9, 283-294 (1996; Zbl 0859.14011)] and a construction of \textit{L. Gruson} and \textit{C. Peskine} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 401-418 (1982; Zbl 0517.14007)] of smooth space curves on a rational quartic surface with a double line in \(\mathbb{P}^3\). The author also proves similar results for curves in \(\mathbb{P}^n\), \(n=3,4,5\), with simple restricted tangent bundle.
scientific article; zbMATH DE number 1066110
scientific article