From the introduction: Several authors studied the normal bundle of spaces curves, but even today our knowledge of this subject is not satisfactory.
The normal bundle of a smooth rational space curve is well-known. D. Eisenbud and A. Van de Ven [Math. Ann. 256, 453-463 (1981; Zbl 0443.14015) and Invent. Math. 67, 89-100 (1982; Zbl 0492.14016)] gave a complete geometric description of the strata associated to the splitting type of the normal bundle of smooth rational space curves. – For \(g\geq 1\), one can stratify the Hilbert scheme of degree \(d\) and genus \(g\) smooth spaces curves \(C\) by the following integer \(s(N_C)\) associated to the normal bundle \(N_C\): \(s(N_C)={1\over 2}\deg N_C-\deg L_{\max}\), where \(L_{\max}\) is a maximal line subbundle of \(N_C\). If \(s(N_C)=s>0\), we say that \(N_C\) is stable with stability degree \(s\). If \(s(N_C)=s<0\), we say that \(N_C\) is unstable with instability degree \(\sigma=-s\). If \(s(N_C)=0\), \(N_C\) is semi-stable non-stable.
Some natural questions arise. For every integer \(s\) such that \(-(d+g-4)\leq s\leq[g/2]\) does exist a smooth space curve \(C\) with \(s(N_C)=s?\) – Let \(N_{d,g}(s)\) be the stratum parametrizing the smooth space curves \(C\) of degree \(d\) and genus \(g\) with \(s(N_C)=s\). Does \(N_{d,g}(s)\) have an irreducible component of the “right” dimension?
For \(g\geq 2\) and a large \(d\) the general degree \(d\) genus \(g\) curve \(C\) has a superstable normal bundle \(N_C\), i.e. \(N_C\) is stable with stability degree \([g/2]\). For \(g=1\), by using the geometric construction of Eisenbud and Van de Ven, K. Hulek and G. Sacchiero [Arch. Math. 40, 61-68 (1983; Zbl 0492.14013)] proved that the degree \(d\) genus 1 general space curve has a semi-stable normal bundle and they found all the instability degrees \(\sigma\) that normal bundles of elliptic curves can admit.
In this paper we find a lot of smooth space curves having an unstable normal bundle. For example, if either \(g\geq 3\) and \(d\geq 4g+2\) or \(g=2\) and \(d\geq 12\), we find a degree \(d\) genus \(g\) smooth curve \(C\) in \(\mathbb{P}^3\) having an unstable normal bundle \(N_C\) with instability degree \(\sigma\), where \(1\leq\sigma\leq d-4\) (theorem 4.3, proposition 4.5, theorem 4.6). – Moreover for \(4g-2\leq\sigma\leq d-4\), we find an irreducible component of the stratum \(N_{d,g}(-\sigma)\) consisting of degree \(d\) genus \(g\) smooth space curves having an unstable normal bundle with instability degree \(\sigma\) of the right dimension, that is \(4d-g+1-2\sigma\) (theorem 4.3). Also for \(g=1\), \(d\geq 7\) and \(3\geq\sigma\leq d-4\), we can find a good irreducible component of the stratum \(N_{d,1}(-\sigma)\) of the right dimension, that is \(4d-2\sigma\). Thus we can calculate the dimension of some strata that K. Hulek and G. Sacchiero found (loc.cit.) (theorem 4.8). We use the geometric construction that D. Eisenbud and A. Van de Ven gave (loc.cit.), i.e. we use a developable ruled surface \(S_L\) containing \(C\) to describe a line subbundle \(L\) of the normal bundle \(N_C\).