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 pencil is a one-dimensional linear system whose general element is an irreducible subvariety that is birational to a smooth variety of Kodaira dimension zero. The following result is proved in [3]. Theorem 1.3. Every Halphen pencil on P is birational to a standard Halphen pencil. Let X be a smooth quartic threefold in P. Then X is not rational, because Bir ( X ) = Aut ( X )