Projectively normal embeddings of an algebraic curve with a net

Let X be a smooth irreducible algebraic curve of genus g. The projective normality of a complete embedding of X is determined by only its quadratic normality in case the embedding is of degree at least g. This means that the complete embedding fails to be projectively normal if and only if it admits an effective divisor which fails to impose independent conditions on quadrics in the embedded projective space. Thus if X admits a net, then it is interesting to compare the conditions for the projective normality of an embedding of X with properties of conic sections of the plane curve given by the net. For such a curve X with a net, we show that the projectively normal embeddings are closely related to properties of conic sections.