Genus of the intersection curve of two rational surface patches

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 m²n² which is generally of genus 2m²n² - 3 2m²n - 3 2n²m + 1. Similarly, two generic tensor product surface patches of parametric degree m₁×m₂ and n₁×n₂ respectively, intersect in a curve of degree 4m₁m₂n₁n₂ and generally of genus 8m₁m₂n₁n₂ - 2m₁m₂(n₁ + n₂) - 2n₁n₂(m₁ + m₂) + 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.