Space curves with the expected postulation
type: Item
instance of: scholarly article
title: Space curves with the expected postulation
zbMATH Open document ID: 1041.14012
DOI: 10.1016/S0022-4049(02)00155-X
author: T. Le Tat
published in: Journal of Pure and Applied Algebra
publication date: 2003-03-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 1878462
zbMATH Keywords: curve with seminatural cohomology
zbMATH Keywords: smoothable curves
zbMATH Keywords: problem of maximal genus
zbMATH Keywords: Horace's method
zbMATH Keywords: Hilbert scheme of space curves
reviewed by: Alberto Dolcetti
MaRDI profile type: Publication
cites work: The maximal rank conjecture for non special curves in \({\mathbb{P}}^ 3\)
cites work: Q3812360
cites work: Hilbert scheme of smooth space curves
cites work: Exemples de courbes de \(P^ 3\) à fibre normal semi-stable, stable
cites work: Construction of space curves with good properties
cites work: Éléments de géométrie algébrique. III: Étude cohomologique des faisceaux cohérents. (Séconde partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Q4143433
cites work: Q3808227
cites work: Q3693625
cites work: Sur la postulation generique des courbes rationnelles
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: Q4838940
description: scientific article; zbMATH DE number 1878462