Brill-Noether loci

Brill-Noether loci Mg,dr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}<^>r_{g,d}$$\end{document} are those subsets of the moduli space Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}_g$$\end{document} 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 <=rho(g,r,d)<= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-g+r+1\le \rho (g,r,d)\le 0$$\end{document} 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.