Primeforms

Primeforms are a set of geometric shapes studied in the field of algebraic geometry. They are defined as the images of the coordinate subspace defined by the vanishing of the first non-zero syzygy of a polynomial under a particular linear map. The concept of primeforms is closely related to the study of minimal free resolutions of modules over polynomial rings.

The definition of primeforms arises from the work of Eisenbud, Reeves, and others. They are particularly relevant when considering the properties of a homogeneous ideal in a polynomial ring. The syzygies of a module provide information about its structure, and the first non-zero syzygy plays a crucial role in determining certain geometric invariants. When this first syzygy is considered as defining a subspace, and this subspace is then mapped linearly, the resulting image is what is termed a primeform.

Primeforms offer a way to study the geometry of algebraic varieties by examining the algebraic structure of their coordinate rings and their associated free resolutions. The properties of primeforms can reveal information about the singularities, the Hilbert function, and other characteristic features of the variety. Research in this area often involves computational algebraic geometry, using software packages to compute resolutions and analyze the resulting primeforms. The study of primeforms is an active area of research with potential applications in various branches of mathematics and theoretical physics.