Brill-Noether loci \({\mathcal {M}}^r_{g,d}\) are those subsets of the moduli space \({\mathcal {M}}_g\) determined by the existence of a linear series of degree d and dimension r. By looking at non-singular curves in a neighborhood of a special chain of elliptic curves, we provide a new proof of the non-emptiness of the Brill-Noether loci when the expected codimension satisfies \(-g+r+1\le \rho (g,r,d)\le 0\) and prove that for a generic point of a component of this locus, the Petri map is onto. As an application, we show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large, substantially generalizing the known result in codimensions 1 and 2. We also provide a new technique for checking that Brill-Noether loci are not included in each other.
