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

Abstract

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