Computing projective equivalences of special algebraic varieties

TL;DR

This paper introduces a unified method for efficiently computing projective equivalences of special algebraic varieties by reducing the problem to finding automorphisms of the projective line, applicable to various classes of varieties.

Contribution

The paper develops a novel approach that simplifies the computation of projective equivalences for specific algebraic varieties using automorphisms of the projective line, unifying multiple cases.

Findings

Effective computation of projective equivalences for rational curves and ruled surfaces.

Method successfully detects affine transformations between planar curves.

Approach identifies symmetries and similarities in algebraic surfaces.

Abstract

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets on the projective line. In particular, we design a unifying approach that finds for two algebraic varieties $X,Y$ from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of candidates for the construction of possible mappings $X\rightarrow Y$. The functionality of the designed method is presented on computing projective equivalences of rational curves, on determining projective equivalences of rational ruled surfaces, on the detection of affine transformations between planar curves, and on computing similarities between two implicitly given algebraic surfaces. When possible, symmetries of given shapes are also discussed as special cases.