Twisted cubics, bis

2001 ◽  
Vol 32 (1) ◽  
pp. 37-44 ◽  
Author(s):  
Israel Vainsencher
Keyword(s):  
Twisted Cubics

scholarly journals The Geometry of Marked Contact Engel Structures

2020 ◽  
Author(s):  
Gianni Manno ◽  
Paweł Nurowski ◽  
Katja Sagerschnig
Keyword(s):  
Local Geometry ◽  
Dimensional Manifold ◽  
Twisted Cubic ◽  
Homogeneous Models ◽  
Local Invariants ◽  
Twisted Cubics ◽  
Cubic Structure ◽  
Contact Distribution ◽  
Engel Structures ◽  
Engel Structure

AbstractA contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$ ( M , C , γ ) is a 5-dimensional manifold $${\mathcal M}$$ M together with a contact distribution $$\mathcal {C}$$ C and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$ γ ⊂ P ( C ) compatible with the conformal symplectic form on $$\mathcal {C}$$ C . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$ G 2 . In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$ σ : M → γ , which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$ γ x . We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$ ( M , C , γ , σ ) , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$ M by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$ γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$ ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

II.—The Irreducible System of Concomitants of Two Double Binary (2, 1) Forms

1926 ◽  
Vol 45 (1) ◽  
pp. 3-13
Author(s):  
W. Saddler
Keyword(s):  
Quadric Surface ◽  
Binary Forms ◽  
Irreducible System ◽  
Simultaneous System ◽  
Twisted Cubics

Little is known of the details of systems of concomitants belonging to double binary forms. The cases of the single ground form of orders (1, 1), (2, 1), (2, 2) respectively, together with the simultaneous system of any number of (1, 1) forms, are the only four cases, which have been published. The following pages establish the simultaneous system of two (2, 1) forms.This system is fundamental for the geometrical treatment of two twisted cubics lying upon a quadric surface and having four common points.

scholarly journals Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

2018 ◽  
Vol 114 ◽  
pp. 85-117 ◽  
Author(s):  
Martí Lahoz ◽  
Manfred Lehn ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
Moduli Space ◽  
Twisted Cubics ◽  
Cubic Fourfold

Surfaces Whose Asymptotic Curves are Twisted Cubics

1938 ◽  
Vol 60 (2) ◽  
pp. 337 ◽  
Author(s):  
E. P. Lane ◽  
M. L. MacQueen
Keyword(s):  
Asymptotic Curves ◽  
Twisted Cubics

Chords of Twisted Cubics

1923 ◽  
Vol s2-21 (1) ◽  
pp. 98-113 ◽  
Author(s):  
E. K. Wakeford
Keyword(s):  
Twisted Cubics

scholarly journals Twisted cubics on cubic fourfolds

2017 ◽  
Vol 2017 (731) ◽  
Author(s):  
Christian Lehn ◽  
Manfred Lehn ◽  
Christoph Sorger ◽  
Duco van Straten
Keyword(s):  
Moduli Space ◽  
Symplectic Manifolds ◽  
Twisted Cubics

AbstractWe construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space

scholarly journals A compactification of the space of twisted cubics

2002 ◽  
Vol 91 (2) ◽  
pp. 221 ◽  
Author(s):  
I. Vainsencher ◽  
F. Xavier
Keyword(s):  
Parameter Space ◽  
Ruled Surfaces ◽  
Segre Varieties ◽  
The Family ◽  
Twisted Cubics ◽  
Nets Of Quadrics ◽  
Smooth Compactification

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in $\boldsymbol P^4$, Segre varieties in $\boldsymbol P^5$. It is suitable for applications of Bott's formula to a few enumerative problems.

Humbert's plane sextics of genus 5

1951 ◽  
Vol 47 (3) ◽  
pp. 483-495 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Twisted Cubics ◽  
Arbitrary Plane ◽  
Pass Through ◽  
Do So

1. In 1894 Humbert encountered a twisted curve C7, of order 7 and genus 5, the locus of points of contact of tangents from a fixed point N0 to those twisted cubics which pass through five fixed points N1, N2, N3, N4, N5. The cubics of this family which touch an arbitrary plane do so at points on a conic, and it was by investigating this complex of conics that Humbert was led to study C7.

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