---
address:
- 'Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 `` 10099 Berlin, Germany'
- 'Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 `` 10099 Berlin, Germany'
author:
- Gavril Farkas
- Angela Ortega
title: 'The maximal rank conjecture and rank two Brill-Noether theory'
---

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Introduction
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The classical Brill-Noether theory of linear series on a curve $[C]\in \cM_g$, which describes the cycles $W^r_d(C):=\{L\in \mbox{Pic}^d(C): h^0(C, L)\geq r+1\}$, is one of the celebrated successes in the theory of algebraic curves. There have been numerous attempts to extend this theory to vector bundles of higher rank, and the subject of this paper is the interplay between Koszul cohomology of line bundles and Brill-Noether phenomena for vector bundles of rank $2$ on curves. Let $\cU_C^s(2, d)$ be the moduli space of stable vector bundles on $C$ of rank $2$ and degree $d$. For each $k\geq 0$, we consider the *Brill-Noether cycle* $$\mathcal{BN}_C(d, k):=\{E\in \cU_C^s(2, d): h^0(C, E)\geq k\}.$$ It is well-known that $\mathcal{BN}_C(d, k)$ has the structure of a determinantal subscheme of $\cU_C^s(2, d)$, and accordingly, each of its irreducible components is of dimension at least equal to the $\emph{Brill-Noether number}$ $\beta_g(d, k):=4g-3-k\bigl(k-d+2g-2\bigr).$ The expectation that 