The space of twisted cubics

Twisted Cubics on Cubic Fourfolds

We construct a new twenty-dimensional family of projective eight-dimensional holomorphically symplectic manifolds: the compactified moduli space M3(Y ) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M3(Y ) → Z(Y ) to a projective eight-dimensional symplectic manifold Z(Y ). The construction is based on results on lin...

The Enumerative Geometry of Plane Cubics. I: Smooth Cubics

We construct a variety of complete plane cubics by a sequence of five blow-ups over P9 . This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.

Ramanujan-nagell Cubics

A well-known result of Beukers [3] on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity |x2 − 2n|. In this paper, we derive an inequality of the shape |x3 − 2n| ≥ x4/3, valid provided x3 6= 2n and (x, n) 6= (5, 7), and then discuss its implications for a variety of Diophantine problems.

Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper in...

Twisted supersymmetry and the topology of theory space