On the normal bundle of space curves in \(\mathbb{P}^3\). (Sur le fibré normal des courbes gauches.) (French) Zbl 0572.14007

Let \(D_s(g)\) (resp. \(D^0_s(g)\), resp. \(D_{ss}(g)\), resp. \(D^0_{ss}(g)\), resp. \(D_P(g))\) be the first integer \(d\) such that there is a smooth connected curve \(C\subset\mathbb{P}^3\) of degree \(d\), genus \(g\), with normal bundle \(N_C\) stable (resp. stable and with \(H^1(C,N_C)=0\), resp. semi-stable, resp. semi-stable and with \(H^1(C,N_C)=0\), resp. with \(H^1(C,N_C(-2))=0)\). The authors announce the following result (which seems to be very useful in many problems about space curves):
(a) For all \(g\geq 2\), \(D^0_s(g)\), \(D_P(g)\) are finite;
(b) if \(g\geq 1\), \(d\geq D^0_s(g)\) (resp. \(D^0_{ss}(g)\), resp. \(D_P(g))\) there is a smooth connected curve \(C\subset\mathbb{P}^3\) with degree \(d\), genus \(g\), with stable normal bundle (resp. semi-stable, resp. with \(H^1(C,N_C(-2))=0)\);
(c) \(D^0_s(g)\leq g+3\);
(d) \(1\leq \lim \sup g^{-2/3}D_{ss}(g)\leq \lim \sup g^{-2/3}D_s(g)\leq \lim \sup g^{-2/3}D^0_{ss}(g)=\lim \sup g^{-2/3}D^0_s(g)\leq \lim \sup D_P(g)\leq (9/8)^{1/3}\);
(f) if \(g\geq 2\), \(d\geq 3g\), there is in \(\mathbb{P}^3\) a smooth curve of genus \(g\), degree \(d\) with normal bundle of degree of stability \([g/2]\).


14F06 Sheaves in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14H10 Families, moduli of curves (algebraic)
14N05 Projective techniques in algebraic geometry