Involutions of Halphen Pencils of Index 2 and Discrete Integrable Systems

Halphen Pencils on Quartic Threefolds

∼= P, where g(x, y, z) = 0 is a homogeneous equation of the curve Z and (λ, μ) ∈ P. Then a general curve in P is birational to an elliptic curve. Remark 1.1. The construction of pencil P can be generalized to the case when C has at most ordinary double points and the points P1, · · · , P9 are not necessarily distinct (see [3]). The pencil P is called a standard Halphen pencil. Definition 1.2. A...

Halphen Pencils on Weighted Fano Threefold Hypersurfaces

We classify all pencils on a general weighted hypersurface of degree ∑ 4 i=1 ai in P(1, a1, a2, a3, a4) whose general members are surfaces of Kodaira dimension zero.

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which...

Discrete Integrable Systems and Geometry

On a Theorem of Halphen and its Application to Integrable Systems

We extend Halphen’s theorem which characterizes the solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n×n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with comple...