Kiemcurves

Kiemcurves are a proposed class of algebraic curves used in theoretical discussions and teaching to illustrate how curve invariants and computational methods interact in algebraic geometry. The term appears in some niche discussions as a generic label for a family of plane algebraic curves.

Definition and construction: A kiemcurve is given by a polynomial equation F(x,y) = 0 over a field K, with F of total degree d. When the curve is smooth, its genus equals (d-1)(d-2)/2. Singular kiemcurves are handled via normalization, which yields a smooth model of genus g, possibly lower than the smooth plane-curve genus.

Properties: Kiemcurves can have various singularity types; their symmetry groups and birational properties are used to classify them. The study often focuses on genus, singularities, and moduli within families defined by parameter values in the coefficients of F.

Computational aspects and examples: For degree 3, a kiemcurve is a plane cubic, which, if smooth, has genus 1 and yields elliptic curves after choosing a rational point. For degree 4, a smooth kiemcurve has genus 3. Algorithms for working with kiemcurves include Gröbner-basis methods, resultants, and numerical algebraic-geometry continuation to approximate points and compute invariants.

Applications: Kiemcurves are used as instructional models in courses on algebraic geometry and computational algebra, and as benchmarking objects for computer-algebra systems. As a flexible framework, they serve to test algorithms for solving polynomial systems and for studying how invariants behave under parameter variation.