ON THE INDEX OF REGULARITY OF NON-SPECIAL SPACE CURVES We work over the complex number field. Fix a curve C c p3, deg(C) = d, pa(C) = g, with d >~ g + 3. C is called of maximal rank if for every integer t > 0 the restriction map rc(t): Hยฐ(P 3, Cp3(t))~ Hยฐ(C, (gc(t)) has maximal rank. If (d, g) # (3, 0) the critical value v of(d, g) (in p3) is the first integer t > 1 such that dt + 1 -g ~< (t + 3)(t + 2)(t + 1)/6. The critical value of (3,0) is 1. If hl((gc(1)) = 0, C has maximal rank if and only ifrc(V -1) is injective and rc(V ) is surjective (Castelnuovo-Mumford's Lemma [14, p. 99]). Let Z'(d, g) be the closure in Hilb(P 3) of the set of smooth, connected curves in p3 with degree d, genus g and non-special hyperplane section. By I-7] the last condition can be dropped. Obviously Z'(d, g) is irreducible. In [5] it was proved that for every (d, g) with d >/g + 3, a general element of Z'(d, g) has maximal rank.
The index of regularity x of the space curve C is the first integer y such that hl(p3,J-c(t)) = 0 for every t >~ y (assuming that C is not projectively normal). By Castelnuovo's theorem (see, e.g., [9], [8]) for any irreducible curve C, deg(C) = d, the index of regularity of C is at most d -2; there are also characterizations of curves with index of regularity d -2 or d -3 ([9], 1,1]). Obviously the index of regularity is at least the critical value (for Z'(d, g)).
In this paper we prove the following result. THEOREM 0.1. Fix integers d, g with d >>. g + 3 >7 3. Let v be the critical value of (d, 9) in p3. Fix an integer n with v <<. n < d -3. Then there is a smooth connected curve X ~ p3, of degree d, genus 9, with non-special hyperplane section, and with index of regularity n.
That is to say, in the non-special range, the index of regularity has Castelnuovo's bound as only constraint: no hole occurs in the set of indices of regularity.
The proof of 0.1 uses heavily the method and proofs of [5] ('la m6thode d'Horace' of [11]). In the first section we use the results of Kleppe's thesis (I-13]) (see [15, Th.l.5]).
Fix integers d, g, v, n as in the statement of 0.1. In Section 3 we define an integer r(n, g) such that d <~ r(n, g) and (r(n, g), g) has critical value n. In the same section (assertion R(n, g), 3.1, 3.2) we prove the existence of a curve Y~ Z'(r(n, g), n) and a line D such that rr(n ) is surjective and card (Yc~ D) = n + 1. Y is