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

TL;DR

This paper investigates the properties of general curve embeddings in projective spaces, specifically focusing on the behavior of quadrics and proving a conjecture related to the maximal rank of these embeddings.

Contribution

It proves a key case of the Maximal Rank Conjecture for quadrics, establishing conditions on the cohomology of ideal sheaves of general curves.

Findings

Either the space of quadrics containing the curve is trivial or its first cohomology vanishes.

The result confirms the Maximal Rank Conjecture in the range of quadrics for general curves.

Provides new insights into the embedding properties of general curves in projective spaces.

Abstract

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).