scholarly journals The Hilbert scheme of space curves of small diameter

2006 ◽  
Vol 56 (5) ◽  
pp. 1297-1335 ◽  
Author(s):  
Jan Kleppe
Keyword(s):  
Hilbert Scheme ◽  
Small Diameter ◽  
Space Curves

scholarly journals Components of the Hilbert scheme of space curves on low-degree smooth surfaces

2015 ◽  
Vol 26 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Jan O. Kleppe ◽  
John C. Ottem
Keyword(s):  
Hilbert Scheme ◽  
Smooth Surface ◽  
Plane Curve ◽  
Picard Number ◽  
Connected Space ◽  
Space Curves ◽  
Cubic Surfaces ◽  
Smooth Component ◽  
Low Degree ◽  
Smooth Plane

We study maximal families W of the Hilbert scheme, H(d, g)sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g)sc and we determine dim W. If s = 4 and W is an irreducible component of H(d, g)sc, then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic (S) is generated by the classes of a line and a smooth plane curve of degree s - 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g)sc, thus making progress in proving a conjecture for such families.

ON THE NUMBER OF IRREDUCIBLE COMPONENTS OF THE HILBERT SCHEME OF SMOOTH SPACE CURVES

1992 ◽  
Vol 03 (06) ◽  
pp. 799-807 ◽  
Author(s):  
PHILIPPE ELLIA ◽  
ANDRÉ HIRSCHOWITZ ◽  
EMILIA MEZZETTI
Keyword(s):  
Hilbert Scheme ◽  
Space Curves ◽  
Irreducible Components ◽  
Smooth Space

scholarly journals The Hilbert Scheme of Buchsbaum space curves

2012 ◽  
Vol 62 (6) ◽  
pp. 2099-2130 ◽  
Author(s):  
Jan Kleppe
Keyword(s):  
Hilbert Scheme ◽  
Space Curves

scholarly journals Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps

2003 ◽  
Vol 171 ◽  
pp. 51-83 ◽  
Author(s):  
Sébastien Guffroy
Keyword(s):  
Hilbert Scheme ◽  
Singular Points ◽  
Plane Curves ◽  
Connected Space ◽  
Space Curves

AbstractSince J.Wahl ([27]), it is known that degree d plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say H, which can be singular. In Wahl’s example, H is singular along a subscheme F but the induced reduced scheme Hred is smooth along F. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of Hred.To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for d < 12 (B.2). Finally we construct our plane curves from the space curves represented by points of this hypersurface (4.7).

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