The space of twisted cubics

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space. Comment: 22 pages. Final version

The Geometry of Marked Contact Engel Structures

A 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.

Twisted cubics on cubic fourfolds

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

QUADRO-QUADRIC CREMONA MAPS AND VARIETIES 3-CONNECTED BY CUBICS: SEMI-SIMPLE PART AND RADICAL

Via the XJC-correspondence proved in [L. Pirio and F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras, submitted] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way, we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi-simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions.

A compactification of the space of twisted cubics

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.