We give a new generalization of the Chudnovsky-Chudnovsky method that
provides upper bounds on the bilinear complexity of multiplication in
monogenous algebras over finite fields through interpolation on algebraic
curves. Two key features of our method is that we allow asymmetric
interpolation, as well as interpolation at arbitrary closed subschemes. This
allows us to fix errors in, improve, and generalize, previous works of
Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-\"Ozbudak.