Genus of the intersection curve of two rational surface patches

Abstract

It is shown that two generic triangular surface patches with no base points and of parametric degree m and n respectively, intersect in a curve of degree m 2 n 2 which is generally of genus 2m 2n 2 − 3 2 m 2n − 3 2 n 2m + 1 . Similarly, two generic tensor product surface patches of parametric degree m 1× m 2 and n 1× n 2 respectively, intersect in a curve of degree 4 m 1 m 2 n 1 n 2 and generally of genus 8 m 1 m 2 n 1 n 2 − 2 m 1 m 2( n 1 + n 2) − 2 n 1 n 2( m 1 + m 2) + 1. For example, two general bicubic patches in general position intersect in a curve of degree 324 and of genus 433. The significance of this genus value lies in the fact that only curves of genus 0 can be expressed parametrically using rational polynomials. Genus and degree equations are also derived for intersection curves involving surface patches with simple base points. A class of surfaces is identified for which any plane section is a rational curve.