Embeddings of general curves in projective spaces: the range of the quadrics

Let $C\subset {\mathbb{P}}^r$ a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either $h^0({\mathbb{P}}^r,{\mathcal{I}}_C(2))=0$ or $h^1(\mathbb {P}^r,\mathcal {I}_C(2)) =0$ (a problem called the Maximal Rank Conjecture in the range of quadrics).