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 $$[g/2]$$.

MSC:

 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