The genus of curves in \(\mathbb{P}^ 4\) verifying certain flag conditions
In \(\mathbb{P}^3\), \textit{G. Halphen} (1883) and \textit{M. Noether} (1883) proposed a bound for the genus of space curves of degree \(d\), not lying on surfaces of degree \(<s\). In higher dimensional spaces \(\mathbb{P}^r\), a similar bound for the genus of curves not lying on surfaes of degree \(<s\) was found by \textit{J. Harris} [``Curves in projective space'', Sémin. Math. Supér. 85, Semin. Sci. OTAN (1982; Zbl 0511.14014)] when \(s\leq 2r-2\) and by \textit{L. Chiantini}, \textit{C. Ciliberto} and \textit{V. Di Gennaro} [Duke Math. J. 70, No. 2, 229-245 (1993; Zbl 0799.14011)] for all \(r,s\) and \(d\gg s\). All the bounds quoted above were proven to be sharp. Curves \(C\) attaining the bounds for \(g\) found in these cited papers lie on surfaces \(S\) of degree \(s\) whose general hyperplane section is a ``Castelnuovo curve'', i.e. a curve of maximal genus in \(\mathbb{P}^{r-1}\). If \(r=4\) it follows that \(S\) (and therefore \(C)\) must lie on a quadric hypersurface, and in general, for \(r\geq 5\), that \(S\) lies on many quadrics. Hence, sticking to the case of \(\mathbb{P}^4\), a natural question to ask is the following problem: Find the maximal genus \(G(d,s,t)\) of an irreducible curve \(C\) in \(\mathbb{P}^4\) of degree \(d\) verifying the following ``flag conditions'': \(C\) does not lie on any surface of degree \(<s\) and on any hypersurface of degree \(<t\). -- In this paper, we give the following answer: Theorem. Let \(C\) be an irreducible reduced, nondegenerate curve of arithmetic genus \(g\) and degree \(d\), in the projective space \(\mathbb{P}^4\) over the complex field. Assume \(C\) not contained on any hypersurface of degree \(<t\) \((t\geq 2)\) and on any surface of degree \(<s\) \((s>t^2-t)\), with \(d>\max (12(s+1)^2, s^3)\). Then the genus \(g\) is bounded by the number \[ G(d,s,t)= {d^2 \over 2s} +{d\over 2} \left[ {s\over t} +t-5-{(t-1-\beta) (1+ \beta)(t-1) \over st} \right]+ \rho+1 \] where \(m,\varepsilon, \alpha,\beta\) are dependent of \(m,s\) and \(t\) only. Furthermore, this bound is sharp and curves \(C\) for which the bound is attained are arithmetically Cohen-Macaulay curves lying in a unique flag of type \(C\subset S\subset T\) where \(T\) is an hypersurface of degree \(t\) and \(S\) is a surface of degree \(s\), whose general hyperplane section \(X\) is a curve of maximal genus in the set of space curves of degree \(s\) not contained on surfaces of degree \(<t\). An easy computation shows that \(G(d,s,2)\) equals the bound found in the authors' previous paper cited above for curves in \(\mathbb{P}^4\) not lying on surfaces of degree \(<s\).
scientific article; zbMATH DE number 866155