On translated rank-2 Brill-Noether loci on regular surfaces

In this note, we carry on the work on the Brill-Noether loci introduced by Costa and Miró-Roig with the infinitesimal study and give a criterion for smoothness. As an illustration, we extend a result of Costa and Miró-Roig (Forum Math 22, 2008, Theorem 3.9) to regular surfaces with H2(X,OX)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2(X,{\mathcal {O}}_X)=0$$\end{document} and bring out a Brill-Noether type description for Qin’s extension spaces (Qin in Manuscr Math 72: 163–180, 1991). We compute the dimension of these loci and find out that it equals the expected dimension, confirming, in this case, the general expectation.