Rationality and Parametrizations of Algebraic Curves under Specializations

TL;DR

This paper investigates how the geometric genus and reducibility of rational algebraic curves change under parameter specialization, providing a partition of parameter space where these properties are invariant and identifying special parameter sets.

Contribution

It introduces a method to partition parameter space for algebraic curves to determine where properties like genus and reducibility are preserved or change under specialization.

Findings

Partition of parameter space where genus and reducibility are invariant

Identification of Zariski-closed sets where genus decreases or curves become reducible

Explicit description for rational curves with special parametrizations

Abstract

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a better description is possible such that the set of parameters where Hilbert's irreducibility theorem does not hold can be isolated, and such that the specialization of the parametrization parametrizes the specialized curve. We conclude the paper by illustrating these results by some concrete applications.