The genus of curves in \(\mathbb{P}^ 4\) verifying certain flag conditions
type: Item
instance of: scholarly article
title: The genus of curves in \(\mathbb{P}^ 4\) verifying certain flag conditions
zbMATH Open document ID: 0866.14016
DOI: 10.1007/BF02567810
author: Luca Chiantini
author: Ciro Ciliberto
author: Vincenzo di Gennaro
published in: Manuscripta Mathematica
publication date: 1996-06-05T00:00:00Z
full work available at URL: https://eudml.org/doc/156115
review text: 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 …
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 866155
zbMATH Keywords: bound for genus of curves in 4-space
MaRDI profile type: Publication
cites work: The genus of projective curves
cites work: On a theorem of Castelnuovo, and the equations defining space curves
cites work: Genre des courbes de l'espace projectif
cites work: Q4694073
OpenAlex ID: W1964966993
description: scientific article; zbMATH DE number 866155