Space curves with the expected postulation
This paper deals with the following conjecture: Let \((d,g)\) be in the \(\text{range}\,FW:= \{(d,g)\mid\exists n\geq 0\) satifying \(0\leq g\leq F(d,n):= nd+1-{n+3 \choose 3}\}\), then there exists a smooth connected curve \(X\subset\mathbb{P}^3\) of degree \(d\) and genus \(g\) with seminatural cohomology (i.e. at most one \(H^i( {\mathcal I}_X(n))\) with \(i=1,2,3\) is nonzero for any \(n)\) and normal bundle verifying the condition \(H^1( {\mathcal N}_X(-1))=0\). It is known that \(\text{range}\,FW\) is the range where there exist curves satisfying \(H^1({\mathcal N}_X(-1))=0\) [see \textit{G. Fløystad}, Math. Ann. 289, No. 1, 33--54 (1991; Zbl 0693.14018)] and that this conjecture solves the so-called problem of maximal genus in \(\text{range}\,A\) [see \textit{R. Hartshorne} in: Algebraic geometry, Proc. Summer Res Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, No. 1, 145--164 (1987; Zbl 0659.14020)]. -- \(\text{Range}\,FW\) is divided in sectors \(F(d,n-1)\leq g\leq F(d,n)\), \(n\geq 1\). The conjecture has been proved in the sector \(n=1\), while a partial answer (i.e. for \(g\leq\frac 43 d-4)\) is known in case \(n=2\) [see \textit{A. Hirschowitz}, Acta Math. 146, 209--230 (1981; Zbl 0475.14027) and \textit{E. Ballico} and \textit{P. Ellia}, Invent. Math. 79, 541--555 (1985; Zbl 0556.14007) and in: Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes Math. 1266, 1--23 (1987; Zbl 0618.14012)]. This paper gives a complete positive answer for the sector \(n=2:d-3\leq g\leq 2d-9\) (theorem 0.1). The proof is based on the inductive argument, known as Horace's method. The author constructs suitable reducible curves, proves that they are smoothable and obtains irreducible components of Hilbert scheme of space curves. The cohomological properties are reached by semicontinuity.
scientific article; zbMATH DE number 1878462
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