Curve subdivision is pivotal in computer graphics for generating smooth geometric objects from control polygons. Traditional methods rely on a global tension parameter, limiting adaptability across diverse curvatures and geometries. This paper introduces a shared learned tension predictor, employing a 140K-parameter network to predict per-edge insertion angles, enabling adaptive curve subdivision across Euclidean, spherical, and hyperbolic geometries. The network incorporates local intrinsic features and a trainable geometry embedding, ensuring specificity without architectural modifications. Theoretical guarantees on structural safety and conditional convergence are provided. Empirical evaluations demonstrate superior performance in bending energy and smoothness compared to fixed-tension baselines, with notable generalisation on out-of-distribution examples.