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We give explicit constructions of all the numerical Campedelli surfaces, i.e. the minimal surfaces of general type with $K^2=2$ and $p_g=0$, whose fundamental group has order 9. There are three families, one with ${\pi }_1^{\text{alg}}={\mathbb{Z}}_9$ and two with ${\pi }_1^{\text{alg}}={\mathbb{Z}}_3^2$. We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with ${\pi }_1^{\text{alg}}={\mathbb{Z}}_9$ and for one of the families of surfaces with ${\pi }_1^{\text{alg}}={\mathbb{Z}}_3^2$ the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces of general...

We take up the study of the Brill-Noether loci $W^r(L,X):=\{\eta \in {\mathrm{Pic}}^0\left(X\right)|h^0(L\otimes \eta )\ge r+1\}$, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm{Pic}\left(X\right)$, and $r\ge 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^0\left(K_D\right)$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$. In the $2$-dimensional case we prove an...